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
Keywords
Examples
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).
Programs
-
Magma
f:=func
; [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]; -
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 *)
-
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
Comments