A353976 Product of the divisors of n whose arithmetic derivative is even.

1, 1, 1, 4, 1, 1, 1, 32, 9, 1, 1, 48, 1, 1, 15, 512, 1, 9, 1, 80, 21, 1, 1, 9216, 25, 1, 9, 112, 1, 15, 1, 16384, 33, 1, 35, 15552, 1, 1, 39, 25600, 1, 21, 1, 176, 135, 1, 1, 7077888, 49, 25, 51, 208, 1, 9, 55, 50176, 57, 1, 1, 864000, 1, 1, 189, 1048576, 65, 33, 1, 272, 69

Offset

1,4

Links

Table of n, a(n) for n=1..69.

Formula

a(n) = Product_{d|n} d^((1+d') mod 2), where d' is the arithmetic derivative of d.

Example

a(8) = 32; the product of the divisors of 8 whose arithmetic derivative is even is 4*8 = 32.

Mathematica

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := Product[k^Mod[d[k] + 1, 2], {k, Divisors[n]}]; Array[a, 60] (* Amiram Eldar, May 13 2022 *)

Prog

(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = vecprod(select(x->!(ad(x) % 2), divisors(n))); \\ Michel Marcus, May 13 2022

Crossrefs

Cf. A003415 (n'), A353975.

Sequence in context: A183104 A183102 A178649 * A363919 A368000 A335502

Adjacent sequences: A353973 A353974 A353975 * A353977 A353978 A353979

Keyword

nonn

Author

Wesley Ivan Hurt, May 12 2022

Status

approved