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

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, 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, 34, 19, 30, 25, 34, 1, 89, 1, 36, 58, 120, 21, 44, 1, 83, 29, 38, 1, 105

Offset: 1

Author

Jaroslav Krizek, Mar 10 2018

Keywords

• nonn

Comments

a(n) >= A054024(n). Conjecture: a(n) = A054024(n) only for the noncomposite numbers A008578.

a(p) = 1 for p = primes.

a(n) = n for numbers: 4, 10, 294, 8388, 612018, 1037952, 3357600, ...

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, ...

From Robert Israel, Mar 11 2018: (Start)

a(p*q) = 3+p+q if p < q are distinct primes and q>3.

a(p^k) = (p^(k+1)-(1+k)*p + k)/(p-1)^2 if p is prime and k >= 0. (End)

Examples

			For n = 4; a(n) = (sigma(1) mod 1 + sigma(2) mod 2 + sigma(4) mod 4) = (0 + 1 + 3) = 4.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A008578, A054024.

Programs

Magma

[(&+[SumOfDivisors(d) mod d: d in Divisors(n)]): n in [1..100]];

Maple

A300657 := n -> add(numtheory:-sigma(d) mod d, d = numtheory:-divisors(n)):
map(A300657, [$1..100]); # Robert Israel, Mar 11 2018

Mathematica

Array[DivisorSum[#, Mod[DivisorSigma[1, #], #] &] &, 72] (* or *)
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 *)

PARI

a(n) = sumdiv(n, d, sigma(d) % d); \\ Michel Marcus, Mar 11 2018

Formula

a(n) = Sum_{d|n} A054024(d).