Arithmetic derivative
The arithmetic derivative of an integer (more specifically, the Lagarias arithmetic derivative) is a function defined for integers, based on prime factorization, by analogy with the product rule for the derivative of a function that is used in mathematical analysis. Accordingly, for natural numbers n, the arithmetic derivative D(n) is defined as follows:
- D(0) = D(1) = 0.
- D(p) = 1 for any prime p.
- D(mn) = D(m)n + mD(n) for any m,n ∈ N. (Leibniz rule for derivatives).
Additionally, for negative integers the arithmetic derivative may be defined as -D(-n) (n < 0).
- Examples
D(2) = 1 and D(3) = 1 (both are prime) so if mn = 2 * 3, D(6) = (1)(3) + (1)(2) = 5.
D(9) = D(3)(3) + D(3)(3) = 6
D(27) = D(3)*9 + D(9)*3 = 9 + 18 = 27
D(30) = D(5)(6) + D(6)(5) = 6 + 5 * 5 = 31.
- Task
Find and show the arithmetic derivatives for -99 through 100.
- Stretch task
Find (the arithmetic derivative of 10^m) then divided by 7, where m is from 1 to 20.
- See also
J
Implementation: <lang J>D=: {{
if. 0 > y do. -D-y
elseif. 2>y do. 0
elseif. 1 p: y do. 1
else. g+({:f)*D g=.*/}:f=. q: y
end.
}}M."0</lang>
Task example:
<lang J> D _99+i.20 10 _75 _77 _1 _272 _24 _49 _34 _96 _20 _123
_1 _140 _32 _45 _22 _124 _1 _43 _108 _176 _1 _71 _18 _80 _55 _39 _1 _156 _1 _59
_26 _72 _1 _61 _18 _192 _51 _33 _1 _92
_1 _31 _22 _92 _16 _81 _1 _56 _20 _45
_14 _112 _1 _25 _39 _48 _1 _41 _1 _68 _16 _21 _1 _60 _12 _19 _14 _80 _1 _31
_1 _32 _27 _15 _10 _44 _1 _13 _10 _24 _1 _21 _1 _32 _8 _9 _1 _16 _1 _7 _6 _12 _1 _5 _1 _4 _1 _1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140</lang>
Also, it seems like it's worth verifying that order of evaluation does not create an ambiguity for the value of D:
<lang J> 15 10 6 + 2 3 5 * D 15 10 6 31 31 31</lang>
Stretch task:
<lang j> (D 10x^1+i.4 5)%7
1 20 300 4000 50000
600000 7000000 80000000 900000000 10000000000
110000000000 1200000000000 13000000000000 140000000000000 1500000000000000
16000000000000000 170000000000000000 1800000000000000000 19000000000000000000 200000000000000000000</lang>
Python
<lang python>from sympy.ntheory import factorint
def D(n):
if n < 0:
return -D(-n)
elif n < 2:
return 0
else:
fdict = factorint(n)
if len(fdict) == 1 and 1 in fdict: # is prime
return 1
return sum([n * e // p for p, e in fdict.items()])
for n in range(-99, 101):
print('{:5}'.format(D(n)), end='\n' if n % 10 == 0 else )
print() for m in range(1, 21):
print('(D for 10**{}) divided by 7 is {}'.format(m, D(10 ** m) // 7))
</lang>
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123
-1 -140 -32 -45 -22 -124 -1 -43 -108 -176
-1 -71 -18 -80 -55 -39 -1 -156 -1 -59
-26 -72 -1 -61 -18 -192 -51 -33 -1 -92
-1 -31 -22 -92 -16 -81 -1 -56 -20 -45
-14 -112 -1 -25 -39 -48 -1 -41 -1 -68
-16 -21 -1 -60 -12 -19 -14 -80 -1 -31
-1 -32 -27 -15 -10 -44 -1 -13 -10 -24
-1 -21 -1 -32 -8 -9 -1 -16 -1 -7
-6 -12 -1 -5 -1 -4 -1 -1 0 0
0 1 1 4 1 5 1 12 6 7
1 16 1 9 8 32 1 21 1 24
10 13 1 44 10 15 27 32 1 31
1 80 14 19 12 60 1 21 16 68
1 41 1 48 39 25 1 112 14 45
20 56 1 81 16 92 22 31 1 92
1 33 51 192 18 61 1 72 26 59
1 156 1 39 55 80 18 71 1 176
108 43 1 124 22 45 32 140 1 123
20 96 34 49 24 272 1 77 75 140
{2: 2}
(D for 10**1) divided by 7 is 1
(D for 10**2) divided by 7 is 20
(D for 10**3) divided by 7 is 300
(D for 10**4) divided by 7 is 4000
(D for 10**5) divided by 7 is 50000
(D for 10**6) divided by 7 is 600000
(D for 10**7) divided by 7 is 7000000
(D for 10**8) divided by 7 is 80000000
(D for 10**9) divided by 7 is 900000000
(D for 10**10) divided by 7 is 10000000000
(D for 10**11) divided by 7 is 110000000000
(D for 10**12) divided by 7 is 1200000000000
(D for 10**13) divided by 7 is 13000000000000
(D for 10**14) divided by 7 is 140000000000000
(D for 10**15) divided by 7 is 1500000000000000
(D for 10**16) divided by 7 is 16000000000000000
(D for 10**17) divided by 7 is 170000000000000000
(D for 10**18) divided by 7 is 1800000000000000000
(D for 10**19) divided by 7 is 19000000000000000000
(D for 10**20) divided by 7 is 200000000000000000000
Raku
<lang perl6>use Prime::Factor;
multi D (0) { 0 } multi D (1) { 0 } multi D ($n where &is-prime) { 1 } multi D ($n where * < 0 ) { -D -$n } multi D ($n) { sum $n.&prime-factors.Bag.map: { $n × .value / .key } }
put (-99 .. 100).map(&D).batch(10)».fmt("%4d").join: "\n";
put ;
put join "\n", (1..20).map: { sprintf "D(10**%-2d) / 7 == %d", $_, D(10**$_) / 7 }</lang>
- Output:
-75 -77 -1 -272 -24 -49 -34 -96 -20 -123 -1 -140 -32 -45 -22 -124 -1 -43 -108 -176 -1 -71 -18 -80 -55 -39 -1 -156 -1 -59 -26 -72 -1 -61 -18 -192 -51 -33 -1 -92 -1 -31 -22 -92 -16 -81 -1 -56 -20 -45 -14 -112 -1 -25 -39 -48 -1 -41 -1 -68 -16 -21 -1 -60 -12 -19 -14 -80 -1 -31 -1 -32 -27 -15 -10 -44 -1 -13 -10 -24 -1 -21 -1 -32 -8 -9 -1 -16 -1 -7 -6 -12 -1 -5 -1 -4 -1 -1 0 0 0 1 1 4 1 5 1 12 6 7 1 16 1 9 8 32 1 21 1 24 10 13 1 44 10 15 27 32 1 31 1 80 14 19 12 60 1 21 16 68 1 41 1 48 39 25 1 112 14 45 20 56 1 81 16 92 22 31 1 92 1 33 51 192 18 61 1 72 26 59 1 156 1 39 55 80 18 71 1 176 108 43 1 124 22 45 32 140 1 123 20 96 34 49 24 272 1 77 75 140 D(10**1 ) / 7 == 1 D(10**2 ) / 7 == 20 D(10**3 ) / 7 == 300 D(10**4 ) / 7 == 4000 D(10**5 ) / 7 == 50000 D(10**6 ) / 7 == 600000 D(10**7 ) / 7 == 7000000 D(10**8 ) / 7 == 80000000 D(10**9 ) / 7 == 900000000 D(10**10) / 7 == 10000000000 D(10**11) / 7 == 110000000000 D(10**12) / 7 == 1200000000000 D(10**13) / 7 == 13000000000000 D(10**14) / 7 == 140000000000000 D(10**15) / 7 == 1500000000000000 D(10**16) / 7 == 16000000000000000 D(10**17) / 7 == 170000000000000000 D(10**18) / 7 == 1800000000000000000 D(10**19) / 7 == 19000000000000000000 D(10**20) / 7 == 200000000000000000000