Powerful numbers
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
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}
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]