The On-Line Encyclopedia of Integer Sequences (OEIS)

A348427 revision #7

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

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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];

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

proposed