A348427 - OEIS

%I #13 Sep 08 2022 08:46:26

%S 10,33,55,145,161,165,253,322,551,649,805,1079,1081,1121,1441,1501,

%T 1513,1633,1653,1711,1771,2353,2755,3237,3401,3403,3713,3841,4321,

%U 4669,4897,5251,5313,5395,5633,5671,6049,6061,6319,6913,7201,7801,8201,8265,8471,10291

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

%C 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.

%e 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).

%e 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).

%t 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 *)

%o (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];

%o (PARI) ad(n) = if (n<1, 0, my(f = factor(n)); n*sum(k=1, #f~, f[k, 2]/f[k, 1])); \\ A003415

%o isok(k) = (k>1) && !isprime(k) && !(sigma(k) % sumdiv(k, d, ad(d))); \\ _Michel Marcus_, Oct 19 2021

%Y Cf. A000203, A003415, A319684.

%K nonn

%O 1,1

%A _Marius A. Burtea_, Oct 18 2021