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A300657 a(n) = Sum_{d|n} sigma(d) mod d.

%I A300657 #24 Apr 06 2025 19:54:24
%S A300657 0,1,1,4,1,2,1,11,5,10,1,9,1,12,11,26,1,9,1,15,13,16,1,28,7,18,18,15,
%T A300657 1,32,1,57,17,22,15,35,1,24,19,32,1,36,1,59,48,28,1,71,9,59,23,67,1,
%U A300657 34,19,30,25,34,1,89,1,36,58,120,21,44,1,83,29,38,1,105
%N A300657 a(n) = Sum_{d|n} sigma(d) mod d.
%C A300657 a(n) >= A054024(n). Conjecture: a(n) = A054024(n) only for the noncomposite numbers A008578.
%C A300657 a(p) = 1 for p = primes.
%C A300657 a(n) = n for numbers: 4, 10, 294, 8388, 612018, 1037952, 3357600, ...
%C A300657 n divides a(n) for numbers: 1, 4, 10, 294, 8388, 218088, 612018, 883386, 1037952, 3357600, ... Corresponding quotients: 0, 1, 1, 1, 1, 2, 1, 2, 1, 1, ...
%C A300657 From _Robert Israel_, Mar 11 2018: (Start)
%C A300657 a(p*q) = 3+p+q if p < q are distinct primes and q>3.
%C A300657 a(p^k) = (p^(k+1)-(1+k)*p + k)/(p-1)^2 if p is prime and k >= 0. (End)
%H A300657 Robert Israel, <a href="/A300657/b300657.txt">Table of n, a(n) for n = 1..10000</a>
%F A300657 a(n) = Sum_{d|n} A054024(d).
%e A300657 For n = 4; a(n) = (sigma(1) mod 1 + sigma(2) mod 2 + sigma(4) mod 4) = (0 + 1 + 3) = 4.
%p A300657 A300657 := n -> add(numtheory:-sigma(d) mod d, d = numtheory:-divisors(n)):
%p A300657 map(A300657, [$1..100]); # _Robert Israel_, Mar 11 2018
%t A300657 Array[DivisorSum[#, Mod[DivisorSigma[1, #], #] &] &, 72] (* or *)
%t A300657 Fold[Function[{a, n}, Append[a, {Total@ Map[a[[#, -1]] &, Most@ Divisors@ n] + #, #} &@ Mod[DivisorSigma[1, n], n]]], {{0, 0}}, Range[2, 72]][[All, 1]] (* _Michael De Vlieger_, Mar 10 2018 *)
%o A300657 (Magma) [(&+[SumOfDivisors(d) mod d: d in Divisors(n)]): n in [1..100]];
%o A300657 (PARI) a(n) = sumdiv(n, d, sigma(d) % d); \\ _Michel Marcus_, Mar 11 2018
%Y A300657 Cf. A008578, A054024.
%K A300657 nonn
%O A300657 1,4
%A A300657 _Jaroslav Krizek_, Mar 10 2018