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
Keywords
Examples
The sum of the nontrivial divisors of 6 whose arithmetic derivative divides 6 is 2 + 3 = 5.
Programs
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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 *)
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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
Formula
a(n) = Sum_{d|n, d'|n, d>1} d, where d' is the arithmetic derivative of d.