A348427 Composite k for which sigma(k) is divisible by the sum of the arithmetic derivatives of the divisors of k.

10, 33, 55, 145, 161, 165, 253, 322, 551, 649, 805, 1079, 1081, 1121, 1441, 1501, 1513, 1633, 1653, 1711, 1771, 2353, 2755, 3237, 3401, 3403, 3713, 3841, 4321, 4669, 4897, 5251, 5313, 5395, 5633, 5671, 6049, 6061, 6319, 6913, 7201, 7801, 8201, 8265, 8471, 10291

Offset

1,1

Comments

Only composite numbers are considered because if p is prime then the sigma(p) = p + 1 is divided by 1' + p' = 0 + 1 = 1 and sigma(p) is divisible of 1.

Links

Table of n, a(n) for n=1..46.

Example

10 is a term because sigma(10) = 1 + 2 + 5 + 10 = 18 is divisible by 1' + 2' + 5' + 10' = 0 + 1 + 1 + 7 = 9 = A319684(10).
33 is a term because sigma(33) = 1 + 3 + 11 + 33 = 48 is divisible by 1' + 3' + 11' + 33' = 0 + 1 + 1 + 14 = 16 = A319684(33).

Mathematica

d[0] = d[1] = 0; d[n_] := n * Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); s[n_] := DivisorSum[n, d[#] &]; Select[Range[10000], CompositeQ[#] && Divisible[DivisorSigma[1, #], s[#]] &] (* Amiram Eldar, Oct 18 2021 *)

Prog

(Magma) f:=func<n |n le 1 select 0 else  n*(&+[Factorisation(n)[i][2]/Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [k:k in [2..10300]|not IsPrime(k) and DivisorSigma(1, k) mod &+[Floor(f(d)): d in Divisors(k)|d ne 1] eq 0];
(PARI) ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415
isok(k) = (k>1) && !isprime(k) && !(sigma(k) % sumdiv(k, d, ad(d))); \\ Michel Marcus, Oct 19 2021

Crossrefs

Cf. A000203, A003415, A319684.

Sequence in context: A067878 A067877 A367031 * A063160 A065149 A299287

Adjacent sequences: A348424 A348425 A348426 * A348428 A348429 A348430

Keyword

nonn

Author

Marius A. Burtea, Oct 18 2021

Status

approved