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A348427
Composite k for which sigma(k) is divisible by the sum of the arithmetic derivatives of the divisors of k.
0
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.
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
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Oct 18 2021
STATUS
approved