A353977 Sum of the nontrivial divisors of n whose arithmetic derivative divides n.

0, 2, 3, 6, 5, 5, 7, 6, 3, 7, 11, 9, 13, 9, 8, 6, 17, 14, 19, 11, 10, 13, 23, 17, 5, 15, 30, 13, 29, 16, 31, 22, 14, 19, 12, 18, 37, 21, 16, 11, 41, 12, 43, 17, 8, 25, 47, 29, 7, 32, 20, 19, 53, 41, 16, 13, 22, 31, 59, 20, 61, 33, 10, 22, 18, 16, 67, 23, 26, 24, 71, 26, 73

Offset

1,2

Links

Table of n, a(n) for n=1..73.

Formula

a(n) = Sum_{d|n, d'|n, d>1} d, where d' is the arithmetic derivative of d.

Example

The sum of the nontrivial divisors of 6 whose arithmetic derivative divides 6 is 2 + 3 = 5.

Mathematica

d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, # &, # > 1 && Divisible[n, d[#]] &]; Array[a, 100] (* Amiram Eldar, May 13 2022 *)

Prog

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

Crossrefs

Cf. A003415, A323880.

Sequence in context: A269374 A137761 A100769 * A023889 A327669 A253413

Adjacent sequences: A353974 A353975 A353976 * A353978 A353979 A353980

Keyword

nonn

Author

Wesley Ivan Hurt, May 12 2022

Status

approved