Powerful numbers
Powerful numbers is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
A k-powerful (or k-full) number is a positive integer n such that for every prime number p dividing n, p^k also divides n.
These are numbers of the form:
2-powerful = a^2 * b^3, for a,b >= 1 3-powerful = a^3 * b^4 * c^5, for a,b,c >= 1 4-powerful = a^4 * b^5 * c^6 * d^7, for a,b,c,d >= 1 ... k-powerful = a^k * b^(k+1) * c^(k+2) *...* ω^(2*k-1), for a,b,c,...,ω >= 1
- Task
Write a function that generates all the k-powerful numbers less than or equal to n.
- For k = 2..10, generate the set of k-powerful numbers <= 10^k and show the first 5 and the last 5 terms, along with the length of the set.
Write a function that counts the number of k-powerful numbers less than or equal to n. (optional)
- For k = 2..10, show the number of k-powerful numbers less than or equal to 10^j, for 0 <= j < k+10.
- See also
Go
Curiously, the 'powerful' function is producing duplicates which the other existing solutions don't seem to suffer from. It's easily dealt with by using a set (rather than a slice) but means that we're unable to take advantage of the counting shortcut - not that it matters as the whole thing still runs in less than 0.4 seconds! <lang go>package main
import (
"fmt" "math" "sort"
)
const adj = 0.0001 // adjustment to ensure f/p square root exact for perfect integer squares
var primes = []uint64{
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
} // seems to be enough
func gcd(x, y uint64) uint64 {
for y != 0 {
x, y = y, x%y
}
return x
}
func isSquareFree(x uint64) bool {
for _, p := range primes {
p2 := p * p
if p2 > x {
break
}
if x%p2 == 0 {
return false
}
}
return true
}
func iroot(x, p uint64) uint64 {
return uint64(math.Pow(float64(x), 1.0/float64(p)) + adj)
}
func ipow(x, p uint64) uint64 {
prod := uint64(1)
for p > 0 {
if p&1 != 0 {
prod *= x
}
p >>= 1
x *= x
}
return prod
}
func powerful(n, k uint64) []uint64 {
set := make(map[uint64]bool)
var f func(m, r uint64) // recursive closure
f = func(m, r uint64) {
if r < k {
set[m] = true
return
}
for v := uint64(1); v <= iroot(n/m, r); v++ {
if r > k {
if !isSquareFree(v) || gcd(m, v) != 1 {
continue
}
}
f(m*ipow(v, r), r-1)
}
}
f(1, (1<<k)-1)
list := make([]uint64, 0, len(set))
for key := range set {
list = append(list, key)
}
sort.Slice(list, func(i, j int) bool {
return list[i] < list[j]
})
return list
}
func main() {
power := uint64(10)
for k := uint64(2); k <= 10; k++ {
power *= 10
a := powerful(power, k)
le := len(a)
h, t := a[0:5], a[le-5:]
fmt.Printf("%d %2d-powerful numbers <= 10^%-2d: %v ... %v\n", le, k, k, h, t)
}
fmt.Println()
for k := uint64(2); k <= 10; k++ {
power := uint64(1)
var counts []int
for j := uint64(0); j < k+10; j++ {
a := powerful(power, k)
counts = append(counts, len(a))
power *= 10
}
j := k + 10
fmt.Printf("Count of %2d-powerful numbers <= 10^j, j in [0, %d): %v\n", k, j, counts)
}
}</lang>
- Output:
14 2-powerful numbers <= 10^2 : [1 4 8 9 16] ... [49 64 72 81 100] 20 3-powerful numbers <= 10^3 : [1 8 16 27 32] ... [625 648 729 864 1000] 25 4-powerful numbers <= 10^4 : [1 16 32 64 81] ... [5184 6561 7776 8192 10000] 32 5-powerful numbers <= 10^5 : [1 32 64 128 243] ... [65536 69984 78125 93312 100000] 38 6-powerful numbers <= 10^6 : [1 64 128 256 512] ... [559872 746496 823543 839808 1000000] 46 7-powerful numbers <= 10^7 : [1 128 256 512 1024] ... [7558272 8388608 8957952 9765625 10000000] 52 8-powerful numbers <= 10^8 : [1 256 512 1024 2048] ... [60466176 67108864 80621568 90699264 100000000] 59 9-powerful numbers <= 10^9 : [1 512 1024 2048 4096] ... [644972544 725594112 816293376 967458816 1000000000] 68 10-powerful numbers <= 10^10: [1 1024 2048 4096 8192] ... [7739670528 8589934592 8707129344 9795520512 10000000000] Count of 2-powerful numbers <= 10^j, j in [0, 12): [1 4 14 54 185 619 2027 6553 21044 67231 214122 680330] Count of 3-powerful numbers <= 10^j, j in [0, 13): [1 2 7 20 51 129 307 713 1645 3721 8348 18589 41136] Count of 4-powerful numbers <= 10^j, j in [0, 14): [1 1 5 11 25 57 117 235 464 906 1741 3312 6236 11654] Count of 5-powerful numbers <= 10^j, j in [0, 15): [1 1 3 8 16 32 63 117 211 375 659 1153 2000 3402 5770] Count of 6-powerful numbers <= 10^j, j in [0, 16): [1 1 2 6 12 21 38 70 121 206 335 551 900 1451 2326 3706] Count of 7-powerful numbers <= 10^j, j in [0, 17): [1 1 1 4 10 16 26 46 77 129 204 318 495 761 1172 1799 2740] Count of 8-powerful numbers <= 10^j, j in [0, 18): [1 1 1 3 8 13 19 32 52 85 135 211 315 467 689 1016 1496 2191] Count of 9-powerful numbers <= 10^j, j in [0, 19): [1 1 1 2 6 11 16 24 38 59 94 145 217 317 453 644 919 1308 1868] Count of 10-powerful numbers <= 10^j, j in [0, 20): [1 1 1 1 5 9 14 21 28 43 68 104 155 227 322 447 621 858 1192 1651]
Julia
<lang julia>using Primes
is_kpowerful(n, k) = all(x -> x[2] >= k, factor(n)) # not used here
is_squarefree(n) = all(x -> x[2] == 1, factor(n)) rootdiv(n, m, r) = Int128(floor(div(n, m)^(1/r) + 0.0000001))
function genkpowerful(n, k)
ret = Int128[]
function inner(m, r)
if r < k
push!(ret, m)
else
for a in 1:rootdiv(n, m, r)
if r <= k || (gcd(a, m) == 1 && is_squarefree(a))
inner(m * Int128(a)^r, r - 1)
end
end
end
end
inner(1, 2 * k - 1)
return unique(sort(Int.(ret)))
end
function kpowerfulcount(n, k)
count = Int128(0)
function inner(m, r)
if r <= k
count += rootdiv(n, m, r)
else
for a in 1:rootdiv(n, m, r)
if gcd(a, m) == 1 && is_squarefree(a)
inner(m * a^r, r - 1)
end
end
end
end
inner(1, 2*k - 1)
return Int(count)
end
for k in 2:10
a = genkpowerful(10^k, k)
len = length(a)
print("The set of $k-powerful numbers between 1 and 10^$k has $len members:\n[")
for i in [1:5 ; len-5:len]
print(a[i], i == 5 ? " ... " : i == len ? "]\n" : ", ")
end
end for k in 2:10
print("The count of $k-powerful numbers from 1 to 10^j for j in 0:$(k+9) is: ",
[kpowerfulcount(Int128(10)^j, k) for j in 0:(k+9)], "\n")
end
</lang>
- Output:
The set of 2-powerful numbers between 1 and 10^2 has 14 members: [1, 4, 8, 9, 16 ... 36, 49, 64, 72, 81, 100] The set of 3-powerful numbers between 1 and 10^3 has 20 members: [1, 8, 16, 27, 32 ... 512, 625, 648, 729, 864, 1000] The set of 4-powerful numbers between 1 and 10^4 has 25 members: [1, 16, 32, 64, 81 ... 4096, 5184, 6561, 7776, 8192, 10000] The set of 5-powerful numbers between 1 and 10^5 has 32 members: [1, 32, 64, 128, 243 ... 62208, 65536, 69984, 78125, 93312, 100000] The set of 6-powerful numbers between 1 and 10^6 has 38 members: [1, 64, 128, 256, 512 ... 531441, 559872, 746496, 823543, 839808, 1000000] The set of 7-powerful numbers between 1 and 10^7 has 46 members: [1, 128, 256, 512, 1024 ... 6718464, 7558272, 8388608, 8957952, 9765625, 10000000] The set of 8-powerful numbers between 1 and 10^8 has 52 members: [1, 256, 512, 1024, 2048 ... 53747712, 60466176, 67108864, 80621568, 90699264, 100000000] The set of 9-powerful numbers between 1 and 10^9 has 59 members: [1, 512, 1024, 2048, 4096 ... 544195584, 644972544, 725594112, 816293376, 967458816, 1000000000] The set of 10-powerful numbers between 1 and 10^10 has 68 members: [1, 1024, 2048, 4096, 8192 ... 6530347008, 7739670528, 8589934592, 8707129344, 9795520512, 10000000000] The count of 2-powerful numbers from 1 to 10^j for j in 0:11 is: [1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330] The count of 3-powerful numbers from 1 to 10^j for j in 0:12 is: [1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136] The count of 4-powerful numbers from 1 to 10^j for j in 0:13 is: [1, 1, 5, 11, 25, 57, 117, 235, 464, 906, 1741, 3312, 6236, 11654] The count of 5-powerful numbers from 1 to 10^j for j in 0:14 is: [1, 1, 3, 8, 16, 32, 63, 117, 211, 375, 659, 1153, 2000, 3402, 5770] The count of 6-powerful numbers from 1 to 10^j for j in 0:15 is: [1, 1, 2, 6, 12, 21, 38, 70, 121, 206, 335, 551, 900, 1451, 2326, 3706] The count of 7-powerful numbers from 1 to 10^j for j in 0:16 is: [1, 1, 1, 4, 10, 16, 26, 46, 77, 129, 204, 318, 495, 761, 1172, 1799, 2740] The count of 8-powerful numbers from 1 to 10^j for j in 0:17 is: [1, 1, 1, 3, 8, 13, 19, 32, 52, 85, 135, 211, 315, 467, 689, 1016, 1496, 2191] The count of 9-powerful numbers from 1 to 10^j for j in 0:18 is: [1, 1, 1, 2, 6, 11, 16, 24, 38, 59, 94, 145, 217, 317, 453, 644, 919, 1308, 1868] The count of 10-powerful numbers from 1 to 10^j for j in 0:19 is: [1, 1, 1, 1, 5, 9, 14, 21, 28, 43, 68, 104, 155, 227, 322, 447, 621, 858, 1192, 1651]
Perl
Generation
<lang perl>use 5.020; use ntheory qw(is_square_free); use experimental qw(signatures); use Math::AnyNum qw(:overload idiv iroot ipow is_coprime);
sub powerful_numbers ($n, $k = 2) {
my @powerful;
sub ($m, $r) {
if ($r < $k) {
push @powerful, $m;
return;
}
for my $v (1 .. iroot(idiv($n, $m), $r)) {
if ($r > $k) {
is_square_free($v) || next;
is_coprime($m, $v) || next;
}
__SUB__->($m * ipow($v, $r), $r - 1);
}
}->(1, 2*$k - 1);
sort { $a <=> $b } @powerful;
}
foreach my $k (2 .. 10) {
my @a = powerful_numbers(10**$k, $k);
my $h = join(', ', @a[0..4]);
my $t = join(', ', @a[$#a-4..$#a]);
printf("For k=%-2d there are %d k-powerful numbers <= 10^k: [%s, ..., %s]\n", $k, scalar(@a), $h, $t);
}</lang>
- Output:
For k=2 there are 14 k-powerful numbers <= 10^k: [1, 4, 8, 9, 16, ..., 49, 64, 72, 81, 100] For k=3 there are 20 k-powerful numbers <= 10^k: [1, 8, 16, 27, 32, ..., 625, 648, 729, 864, 1000] For k=4 there are 25 k-powerful numbers <= 10^k: [1, 16, 32, 64, 81, ..., 5184, 6561, 7776, 8192, 10000] For k=5 there are 32 k-powerful numbers <= 10^k: [1, 32, 64, 128, 243, ..., 65536, 69984, 78125, 93312, 100000] For k=6 there are 38 k-powerful numbers <= 10^k: [1, 64, 128, 256, 512, ..., 559872, 746496, 823543, 839808, 1000000] For k=7 there are 46 k-powerful numbers <= 10^k: [1, 128, 256, 512, 1024, ..., 7558272, 8388608, 8957952, 9765625, 10000000] For k=8 there are 52 k-powerful numbers <= 10^k: [1, 256, 512, 1024, 2048, ..., 60466176, 67108864, 80621568, 90699264, 100000000] For k=9 there are 59 k-powerful numbers <= 10^k: [1, 512, 1024, 2048, 4096, ..., 644972544, 725594112, 816293376, 967458816, 1000000000] For k=10 there are 68 k-powerful numbers <= 10^k: [1, 1024, 2048, 4096, 8192, ..., 7739670528, 8589934592, 8707129344, 9795520512, 10000000000]
Counting
<lang perl>use 5.020; use ntheory qw(is_square_free); use experimental qw(signatures); use Math::AnyNum qw(:overload idiv iroot ipow is_coprime);
sub powerful_count ($n, $k = 2) {
my $count = 0;
sub ($m, $r) {
if ($r <= $k) {
$count += iroot(idiv($n, $m), $r);
return;
}
for my $v (1 .. iroot(idiv($n, $m), $r)) {
is_square_free($v) || next;
is_coprime($m, $v) || next;
__SUB__->($m * ipow($v, $r), $r - 1);
}
}->(1, 2*$k - 1);
return $count;
}
foreach my $k (2 .. 10) {
printf("Number of %2d-powerful <= 10^j for 0 <= j < %d: {%s}\n", $k, $k+10,
join(', ', map { powerful_count(ipow(10, $_), $k) } 0..($k+10-1)));
}</lang>
- Output:
Number of 2-powerful <= 10^j for 0 <= j < 12: {1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330}
Number of 3-powerful <= 10^j for 0 <= j < 13: {1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136}
Number of 4-powerful <= 10^j for 0 <= j < 14: {1, 1, 5, 11, 25, 57, 117, 235, 464, 906, 1741, 3312, 6236, 11654}
Number of 5-powerful <= 10^j for 0 <= j < 15: {1, 1, 3, 8, 16, 32, 63, 117, 211, 375, 659, 1153, 2000, 3402, 5770}
Number of 6-powerful <= 10^j for 0 <= j < 16: {1, 1, 2, 6, 12, 21, 38, 70, 121, 206, 335, 551, 900, 1451, 2326, 3706}
Number of 7-powerful <= 10^j for 0 <= j < 17: {1, 1, 1, 4, 10, 16, 26, 46, 77, 129, 204, 318, 495, 761, 1172, 1799, 2740}
Number of 8-powerful <= 10^j for 0 <= j < 18: {1, 1, 1, 3, 8, 13, 19, 32, 52, 85, 135, 211, 315, 467, 689, 1016, 1496, 2191}
Number of 9-powerful <= 10^j for 0 <= j < 19: {1, 1, 1, 2, 6, 11, 16, 24, 38, 59, 94, 145, 217, 317, 453, 644, 919, 1308, 1868}
Number of 10-powerful <= 10^j for 0 <= j < 20: {1, 1, 1, 1, 5, 9, 14, 21, 28, 43, 68, 104, 155, 227, 322, 447, 621, 858, 1192, 1651}
Perl 6
Perl 6 has no handy pre-made nth integer root routine so has the same problem as Go with needing a slight "fudge factor" in the nth root calculation.
<lang perl6>sub super (\x) { x.trans([<0123456789>.comb] => [<⁰¹²³⁴⁵⁶⁷⁸⁹>.comb]) }
sub is-square-free (Int \n) {
constant @p = ^100 .map: { next unless .is-prime; .² };
for @p -> \p { return False if n %% p }
True
}
sub powerfuls (\n, \k, \enumerate = False) {
my @powerful;
p(1, 2*k - 1);
sub p (\m, \r) {
if r < k {
enumerate ?? @powerful.push(m) !! ++@powerful[m - 1 ?? (m - 1).chars !! 0];
return
}
for 1 .. ((n / m) ** (1/r) + .0001).Int -> \v {
if r > k {
next unless is-square-free(v);
next unless m gcd v == 1;
}
p(m * v ** r, r - 1)
}
}
@powerful;
}
put "Count and first and last five enumerated n-powerful numbers in 10ⁿ:"; for 2..10 -> \k {
my @powerful = sort powerfuls(10**k, k, True);
printf "%2d %2s-powerful numbers <= 10%-2s: %s ... %s\n", +@powerful, k, super(k),
@powerful.head(5).join(", "), @powerful.tail(5).join(", ");
}
put "\nCounts in each order of magnitude:"; my $top = 9; for 2..10 -> \k {
printf "%2s-powerful numbers <= 10ⁿ (where 0 <= n <= %d): ", k, $top+k; quietly say join ', ', [\+] powerfuls(10**($top + k), k);
}</lang>
- Output:
Count and first and last five enumerated n-powerful numbers in 10ⁿ: 14 2-powerful numbers <= 10² : 1, 4, 8, 9, 16 ... 49, 64, 72, 81, 100 20 3-powerful numbers <= 10³ : 1, 8, 16, 27, 32 ... 625, 648, 729, 864, 1000 25 4-powerful numbers <= 10⁴ : 1, 16, 32, 64, 81 ... 5184, 6561, 7776, 8192, 10000 32 5-powerful numbers <= 10⁵ : 1, 32, 64, 128, 243 ... 65536, 69984, 78125, 93312, 100000 38 6-powerful numbers <= 10⁶ : 1, 64, 128, 256, 512 ... 559872, 746496, 823543, 839808, 1000000 46 7-powerful numbers <= 10⁷ : 1, 128, 256, 512, 1024 ... 7558272, 8388608, 8957952, 9765625, 10000000 52 8-powerful numbers <= 10⁸ : 1, 256, 512, 1024, 2048 ... 60466176, 67108864, 80621568, 90699264, 100000000 59 9-powerful numbers <= 10⁹ : 1, 512, 1024, 2048, 4096 ... 644972544, 725594112, 816293376, 967458816, 1000000000 68 10-powerful numbers <= 10¹⁰: 1, 1024, 2048, 4096, 8192 ... 7739670528, 8589934592, 8707129344, 9795520512, 10000000000 Counts in each order of magnitude: 2-powerful numbers <= 10ⁿ (where 0 <= n <= 11): 1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330 3-powerful numbers <= 10ⁿ (where 0 <= n <= 12): 1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136 4-powerful numbers <= 10ⁿ (where 0 <= n <= 13): 1, 1, 5, 11, 25, 57, 117, 235, 464, 906, 1741, 3312, 6236, 11654 5-powerful numbers <= 10ⁿ (where 0 <= n <= 14): 1, 1, 3, 8, 16, 32, 63, 117, 211, 375, 659, 1153, 2000, 3402, 5770 6-powerful numbers <= 10ⁿ (where 0 <= n <= 15): 1, 1, 2, 6, 12, 21, 38, 70, 121, 206, 335, 551, 900, 1451, 2326, 3706 7-powerful numbers <= 10ⁿ (where 0 <= n <= 16): 1, 1, 1, 4, 10, 16, 26, 46, 77, 129, 204, 318, 495, 761, 1172, 1799, 2740 8-powerful numbers <= 10ⁿ (where 0 <= n <= 17): 1, 1, 1, 3, 8, 13, 19, 32, 52, 85, 135, 211, 315, 467, 689, 1016, 1496, 2191 9-powerful numbers <= 10ⁿ (where 0 <= n <= 18): 1, 1, 1, 2, 6, 11, 16, 24, 38, 59, 94, 145, 217, 317, 453, 644, 919, 1308, 1868 10-powerful numbers <= 10ⁿ (where 0 <= n <= 19): 1, 1, 1, 1, 5, 9, 14, 21, 28, 43, 68, 104, 155, 227, 322, 447, 621, 858, 1192, 1651
Phix
priority queue
Tried a slightly different approach, but it gets 7/10 at best on performance. <lang Phix>include mpfr.e
integer pq = NULL -- data is the prime powers, eg {2,2} for 2^2*3^2
-- priority is the mpz of that, eg/ie 36
procedure padd(mpz n, np, sequence s)
if pq=NULL then
pq = pq_new(MIN_HEAP,routine_id("mpz_cmp"))
end if
if mpz_cmp(np,n)<=0 then
pq_add({s,mpz_init_set(np)},pq)
end if
end procedure
procedure add_next(integer k, mpz n, p, sequence s)
mpz np = mpz_init()
integer l = length(s)
for i=1 to l do
integer ki = iff(s[i]=0?k:1)
mpz_ui_pow_ui(np,get_prime(i),ki)
mpz_mul(np,np,p)
s[i] += ki
padd(n,np,s)
s[i] -= ki
end for
mpz_ui_pow_ui(np,get_prime(l+1),k)
if l>0 and s[l]=k and s[1..l-1]==repeat(0,l-1) then
padd(n,np,0&s)
elsif s={} then
mpz_mul(np,np,p)
padd(n,np,s&k)
end if
end procedure
function powerful(integer k, mpz n)
mpz prevp = mpz_init(1)
sequence res = {prevp}
add_next(k,n,prevp,{})
while pq_size(pq) do
{sequence s, mpz p} = pq_pop(pq)
if mpz_cmp(p,prevp)!=0 then
res &= p
add_next(k,n,p,s)
prevp = p
end if
end while
return res
end function
mpz p = mpz_init(10) for k=2 to 10 do
mpz_mul_si(p,p,10)
sequence s = powerful(k,p)
integer l = length(s)
for i=1 to 5 do
s[i] = mpz_get_str(s[i])
s[-i] = mpz_get_str(s[-i])
end for
s[6..-6] = {"..."}
s = join(s,",")
printf(1,"%d %2d-powerful numbers <= 10^%-2d : %s\n",{l,k,k,s})
end for
puts(1,"\n") --Not really fast enough, so only doing to k+5..k+8 for the first 4, but k+9 for last 5 for k=2 to 10 do
sequence counts = {}
integer lim = k+min(k+3,9)
mpz_set_si(p,1)
for j=0 to lim do
printf(1,"counting (%d/%d)...\r",{j,lim})
counts &= length(powerful(k,p))
mpz_mul_si(p,p,10)
end for
printf(1,"Counts of %2d-powerful numbers <= 10^(0..%d) : %v\n",{k,lim,counts})
end for</lang>
- Output:
14 2-powerful numbers <= 10^2 : 1,4,8,9,16,...,49,64,72,81,100
20 3-powerful numbers <= 10^3 : 1,8,16,27,32,...,625,648,729,864,1000
25 4-powerful numbers <= 10^4 : 1,16,32,64,81,...,5184,6561,7776,8192,10000
32 5-powerful numbers <= 10^5 : 1,32,64,128,243,...,65536,69984,78125,93312,100000
38 6-powerful numbers <= 10^6 : 1,64,128,256,512,...,559872,746496,823543,839808,1000000
46 7-powerful numbers <= 10^7 : 1,128,256,512,1024,...,7558272,8388608,8957952,9765625,10000000
52 8-powerful numbers <= 10^8 : 1,256,512,1024,2048,...,60466176,67108864,80621568,90699264,100000000
59 9-powerful numbers <= 10^9 : 1,512,1024,2048,4096,...,644972544,725594112,816293376,967458816,1000000000
68 10-powerful numbers <= 10^10 : 1,1024,2048,4096,8192,...,7739670528,8589934592,8707129344,9795520512,10000000000
Counts of 2-powerful numbers <= 10^(0..7) : {1,4,14,54,185,619,2027,6553}
Counts of 3-powerful numbers <= 10^(0..9) : {1,2,7,20,51,129,307,713,1645,3721}
Counts of 4-powerful numbers <= 10^(0..11) : {1,1,5,11,25,57,117,235,464,906,1741,3312}
Counts of 5-powerful numbers <= 10^(0..13) : {1,1,3,8,16,32,63,117,211,375,659,1153,2000,3402}
Counts of 6-powerful numbers <= 10^(0..15) : {1,1,2,6,12,21,38,70,121,206,335,551,900,1451,2326,3706}
Counts of 7-powerful numbers <= 10^(0..16) : {1,1,1,4,10,16,26,46,77,129,204,318,495,761,1172,1799,2740}
Counts of 8-powerful numbers <= 10^(0..17) : {1,1,1,3,8,13,19,32,52,85,135,211,315,467,689,1016,1496,2191}
Counts of 9-powerful numbers <= 10^(0..18) : {1,1,1,2,6,11,16,24,38,59,94,145,217,317,453,644,919,1308,1868}
Counts of 10-powerful numbers <= 10^(0..19) : {1,1,1,1,5,9,14,21,28,43,68,104,155,227,322,447,621,858,1192,1651}
translation
Significantly faster. The last few counts were wrong when using native ints/atoms, so I went gmp, which means it also works on 32-bit. <lang Phix>include mpfr.e
function powerful(mpz n, integer k, sequence res={}, mpz m=NULL, integer r=0)
if m=NULL then
m = mpz_init(1)
r = 2*k-1
end if
if r<k then
res = append(res,iff(mpz_fits_atom(m)?mpz_get_atom(m):mpz_get_str(m)))
else
mpz t = mpz_init()
mpz_fdiv_q(t, n, m) -- t := floor(n/m)
{} = mpz_root(t,t,r) -- t = floor(rth root of t)
if not mpz_fits_integer(t) then ?9/0 end if
integer lim = mpz_get_integer(t)
for v=1 to lim do
if r<=k
or (square_free(v) and mpz_gcd_ui(NULL,m,v)=1) then
mpz_ui_pow_ui(t,v,r) -- t:= v^r
mpz_mul(t,t,m) -- t*= m
res = powerful(n, k, res, t, r-1)
end if
end for
end if
return res
end function
function powercount(mpz n, integer k, mpz m=NULL, integer r=0)
atom res = 0
if m=NULL then
m = mpz_init(1)
r = 2*k-1
end if
mpz t = mpz_init()
mpz_fdiv_q(t, n, m) -- t := floor(n/m)
{} = mpz_root(t,t,r) -- t = floor(rth root of t)
if not mpz_fits_integer(t) then ?9/0 end if
integer lim = mpz_get_integer(t)
if r<=k then
res += lim
else
for v=1 to lim do
if square_free(v) and mpz_gcd_ui(NULL,m,v)=1 then
mpz_ui_pow_ui(t,v,r) -- t:= v^r
mpz_mul(t,t,m) -- t*= m
res += powercount(n, k, t, r-1)
end if
end for
end if
return res
end function
atom t0 = time() mpz p = mpz_init(10) for k=2 to 10 do
mpz_mul_si(p,p,10)
sequence s = sort(powerful(p, k))
integer l = length(s)
s[6..-6] = {"..."}
s = ppf(s,{pp_FltFmt,"%d",pp_StrFmt,-1,pp_IntCh,false,pp_Maxlen,100})
printf(1,"%d %2d-powerful numbers <= 10^%-2d : %s\n",{l,k,k,s})
end for puts(1,"\n") for k=2 to 10 do
mpz_set_si(p,1)
sequence counts = {}
for j=0 to k+9 do
counts = append(counts, powercount(p, k))
mpz_mul_si(p,p,10)
end for
printf(1,"Counts of %2d-powerful numbers <= 10^(0..%d) : %v\n",{k,k+9,counts})
end for ?elapsed(time()-t0)</lang>
- Output:
14 2-powerful numbers <= 10^2 : {1,4,8,9,16, ..., 49,64,72,81,100}
20 3-powerful numbers <= 10^3 : {1,8,16,27,32, ..., 625,648,729,864,1000}
25 4-powerful numbers <= 10^4 : {1,16,32,64,81, ..., 5184,6561,7776,8192,10000}
32 5-powerful numbers <= 10^5 : {1,32,64,128,243, ..., 65536,69984,78125,93312,100000}
38 6-powerful numbers <= 10^6 : {1,64,128,256,512, ..., 559872,746496,823543,839808,1000000}
46 7-powerful numbers <= 10^7 : {1,128,256,512,1024, ..., 7558272,8388608,8957952,9765625,10000000}
52 8-powerful numbers <= 10^8 : {1,256,512,1024,2048, ..., 60466176,67108864,80621568,90699264,100000000}
59 9-powerful numbers <= 10^9 : {1,512,1024,2048,4096, ..., 644972544,725594112,816293376,967458816,1000000000}
68 10-powerful numbers <= 10^10 : {1,1024,2048,4096,8192, ..., 7739670528,8589934592,8707129344,9795520512,10000000000}
Counts of 2-powerful numbers <= 10^(0..11) : {1,4,14,54,185,619,2027,6553,21044,67231,214122,680330}
Counts of 3-powerful numbers <= 10^(0..12) : {1,2,7,20,51,129,307,713,1645,3721,8348,18589,41136}
Counts of 4-powerful numbers <= 10^(0..13) : {1,1,5,11,25,57,117,235,464,906,1741,3312,6236,11654}
Counts of 5-powerful numbers <= 10^(0..14) : {1,1,3,8,16,32,63,117,211,375,659,1153,2000,3402,5770}
Counts of 6-powerful numbers <= 10^(0..15) : {1,1,2,6,12,21,38,70,121,206,335,551,900,1451,2326,3706}
Counts of 7-powerful numbers <= 10^(0..16) : {1,1,1,4,10,16,26,46,77,129,204,318,495,761,1172,1799,2740}
Counts of 8-powerful numbers <= 10^(0..17) : {1,1,1,3,8,13,19,32,52,85,135,211,315,467,689,1016,1496,2191}
Counts of 9-powerful numbers <= 10^(0..18) : {1,1,1,2,6,11,16,24,38,59,94,145,217,317,453,644,919,1308,1868}
Counts of 10-powerful numbers <= 10^(0..19) : {1,1,1,1,5,9,14,21,28,43,68,104,155,227,322,447,621,858,1192,1651}
"0.8s"
Sidef
Generation
<lang ruby>func powerful(n, k=2) {
var list = []
func (m,r) {
if (r < k) {
list << m
return nil
}
for a in (1 .. iroot(idiv(n,m), r)) {
if (r > k) {
a.is_coprime(m) || next
a.is_squarefree || next
}
__FUNC__(m * a**r, r-1)
}
}(1, 2*k - 1)
return list.sort
}
for k in (2..10) {
var a = powerful(10**k, k)
var h = a.head(5).join(', ')
var t = a.tail(5).join(', ')
printf("For k=%-2d there are %d k-powerful numbers <= 10^k: [%s, ..., %s]\n", k, a.len, h, t)
}</lang>
- Output:
For k=2 there are 14 k-powerful numbers <= 10^k: [1, 4, 8, 9, 16, ..., 49, 64, 72, 81, 100] For k=3 there are 20 k-powerful numbers <= 10^k: [1, 8, 16, 27, 32, ..., 625, 648, 729, 864, 1000] For k=4 there are 25 k-powerful numbers <= 10^k: [1, 16, 32, 64, 81, ..., 5184, 6561, 7776, 8192, 10000] For k=5 there are 32 k-powerful numbers <= 10^k: [1, 32, 64, 128, 243, ..., 65536, 69984, 78125, 93312, 100000] For k=6 there are 38 k-powerful numbers <= 10^k: [1, 64, 128, 256, 512, ..., 559872, 746496, 823543, 839808, 1000000] For k=7 there are 46 k-powerful numbers <= 10^k: [1, 128, 256, 512, 1024, ..., 7558272, 8388608, 8957952, 9765625, 10000000] For k=8 there are 52 k-powerful numbers <= 10^k: [1, 256, 512, 1024, 2048, ..., 60466176, 67108864, 80621568, 90699264, 100000000] For k=9 there are 59 k-powerful numbers <= 10^k: [1, 512, 1024, 2048, 4096, ..., 644972544, 725594112, 816293376, 967458816, 1000000000] For k=10 there are 68 k-powerful numbers <= 10^k: [1, 1024, 2048, 4096, 8192, ..., 7739670528, 8589934592, 8707129344, 9795520512, 10000000000]
Counting
<lang ruby>func powerful_count(n, k=2) {
var count = 0
func (m,r) {
if (r <= k) {
count += iroot(idiv(n,m), r)
return nil
}
for a in (1 .. iroot(idiv(n,m), r)) {
a.is_coprime(m) || next
a.is_squarefree || next
__FUNC__(m * a**r, r-1)
}
}(1, 2*k - 1)
return count
}
for k in (2..10) {
var a = (k+10).of {|j| powerful_count(10**j, k) }
printf("Number of %2d-powerful numbers <= 10^j, for 0 <= j < #{k+10}: %s\n", k, a)
}</lang>
- Output:
Number of 2-powerful numbers <= 10^j, for 0 <= j < 12: [1, 4, 14, 54, 185, 619, 2027, 6553, 21044, 67231, 214122, 680330] Number of 3-powerful numbers <= 10^j, for 0 <= j < 13: [1, 2, 7, 20, 51, 129, 307, 713, 1645, 3721, 8348, 18589, 41136] Number of 4-powerful numbers <= 10^j, for 0 <= j < 14: [1, 1, 5, 11, 25, 57, 117, 235, 464, 906, 1741, 3312, 6236, 11654] Number of 5-powerful numbers <= 10^j, for 0 <= j < 15: [1, 1, 3, 8, 16, 32, 63, 117, 211, 375, 659, 1153, 2000, 3402, 5770] Number of 6-powerful numbers <= 10^j, for 0 <= j < 16: [1, 1, 2, 6, 12, 21, 38, 70, 121, 206, 335, 551, 900, 1451, 2326, 3706] Number of 7-powerful numbers <= 10^j, for 0 <= j < 17: [1, 1, 1, 4, 10, 16, 26, 46, 77, 129, 204, 318, 495, 761, 1172, 1799, 2740] Number of 8-powerful numbers <= 10^j, for 0 <= j < 18: [1, 1, 1, 3, 8, 13, 19, 32, 52, 85, 135, 211, 315, 467, 689, 1016, 1496, 2191] Number of 9-powerful numbers <= 10^j, for 0 <= j < 19: [1, 1, 1, 2, 6, 11, 16, 24, 38, 59, 94, 145, 217, 317, 453, 644, 919, 1308, 1868] Number of 10-powerful numbers <= 10^j, for 0 <= j < 20: [1, 1, 1, 1, 5, 9, 14, 21, 28, 43, 68, 104, 155, 227, 322, 447, 621, 858, 1192, 1651]
zkl
<lang zkl>fcn isSquareFree(n){
var [const] primes2=T(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,) // seems to be enough
.apply("pow",2);
foreach p2 in (primes2){
if(p2>n) break;
if(n%p2 == 0) return(False);
}
True
}
const adj=0.0001; // adjustment to ensure f/r square root exact for perfect integer squares fcn iroot(n,r){ (n.toFloat().pow(1.0/r) + adj).toInt() }
fcn powerfuls(n,k){
fcn(m,r, n,k,powerful){ // recursive closure
if(r<k){ powerful[m]=True; return(); }
foreach v in ([1..iroot(n/m, r)]){
if(r>k){ if(not isSquareFree(v) or m.gcd(v)!=1) continue; }
self.fcn(m*v.pow(r), r-1, n,k,powerful)
}
}(1,(2).pow(k)-1, n,k, powerful:=Dictionary());
powerful.keys.apply("toInt").sort();
}</lang> <lang zkl>println("Count, first and last five enumerated n-powerful numbers in 10\u207f:"); foreach k in ([2..10]){
ps:=powerfuls((10).pow(k),k);
println("%2d: %s ... %s".fmt(ps.len(),ps[0,5].concat(" "),ps.tail(5).concat(" ")));
}</lang>
- Output:
Count, first and last five enumerated n-powerful numbers in 10ⁿ: 14: 1 4 8 9 16 ... 49 64 72 81 100 20: 1 8 16 27 32 ... 625 648 729 864 1000 25: 1 16 32 64 81 ... 5184 6561 7776 8192 10000 32: 1 32 64 128 243 ... 65536 69984 78125 93312 100000 38: 1 64 128 256 512 ... 559872 746496 823543 839808 1000000 46: 1 128 256 512 1024 ... 7558272 8388608 8957952 9765625 10000000 52: 1 256 512 1024 2048 ... 60466176 67108864 80621568 90699264 100000000 59: 1 512 1024 2048 4096 ... 644972544 725594112 816293376 967458816 1000000000 68: 1 1024 2048 4096 8192 ... 7739670528 8589934592 8707129344 9795520512 10000000000
Overflows a 64 bit int at the very end, which results in a count of zero. <lang zkl>fcn powerfulsN(n,k){ // count
fcn(m,r, n,k,count){ // recursive closure
if(r<=k){ count.incN(iroot(n/m, r)); return() }
foreach v in ([1..iroot(n/m, r)]){
if(not isSquareFree(v) or m.gcd(v)!=1) continue;
self.fcn(m*v.pow(r), r-1, n,k,count)
} }(1,2*k - 1, n,k, count:=Ref(0)); count.value
}</lang> <lang zkl>println("Counts in each order of magnitude:"); foreach k in ([2..10]){
print("%2s-powerful numbers <= 10\u207f (n in [0..%d)): ".fmt(k, 10+k));
ps:=[0 .. k+10 -1].apply('wrap(n){ powerfulsN((10).pow(n), k) });
println(ps.concat(" "));
}</lang>
- Output:
Counts in each order of magnitude: 2-powerful numbers <= 10ⁿ (n in [0..12)): 1 4 14 54 185 619 2027 6553 21044 67231 214122 680330 3-powerful numbers <= 10ⁿ (n in [0..13)): 1 2 7 20 51 129 307 713 1645 3721 8348 18589 41136 4-powerful numbers <= 10ⁿ (n in [0..14)): 1 1 5 11 25 57 117 235 464 906 1741 3312 6236 11654 5-powerful numbers <= 10ⁿ (n in [0..15)): 1 1 3 8 16 32 63 117 211 375 659 1153 2000 3402 5770 6-powerful numbers <= 10ⁿ (n in [0..16)): 1 1 2 6 12 21 38 70 121 206 335 551 900 1451 2326 3706 7-powerful numbers <= 10ⁿ (n in [0..17)): 1 1 1 4 10 16 26 46 77 129 204 318 495 761 1172 1799 2740 8-powerful numbers <= 10ⁿ (n in [0..18)): 1 1 1 3 8 13 19 32 52 85 135 211 315 467 689 1016 1496 2191 9-powerful numbers <= 10ⁿ (n in [0..19)): 1 1 1 2 6 11 16 24 38 59 94 145 217 317 453 644 919 1308 1868 10-powerful numbers <= 10ⁿ (n in [0..20)): 1 1 1 1 5 9 14 21 28 43 68 104 155 227 322 447 621 858 1192 0