A318996 a(n) = Sum_{d|n} (sigma(n) mod d).

0, 1, 1, 4, 1, 0, 1, 11, 5, 11, 1, 9, 1, 13, 13, 26, 1, 10, 1, 8, 17, 17, 1, 16, 7, 19, 18, 0, 1, 28, 1, 57, 19, 23, 22, 34, 1, 25, 23, 24, 1, 41, 1, 65, 45, 29, 1, 57, 9, 68, 25, 75, 1, 39, 25, 25, 29, 35, 1, 88, 1, 37, 74, 120, 29, 37, 1, 91, 31, 24, 1, 103

Offset: 1

Author

Jaroslav Krizek, Sep 07 2018

Keywords

• nonn

Examples

			For n = 4; a(4) = (7 mod 1) + (7 mod 2) + (7 mod 4) = 0 + 1 + 3 = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..65537
• Carlos Rivera, Puzzle 1065. A larger integer than 45 such that ..., The Prime Puzzles and Problems Connection.

Crossrefs

Cf. A000005, A000040, A000203, A007691, A008578, A300657.

Programs

Magma

[&+[SumOfDivisors(n) mod d: d in Divisors(n)] : n in [1..1000]]

Mathematica

a[n_] := Block[{s = DivisorSigma[1, n]}, DivisorSum[n, Mod[s, #] &]]; Array[a, 72] (* Giovanni Resta, Sep 07 2018 *)

PARI

a(n) = my(sn = sigma(n)); sumdiv(n, d, sn % d); \\ Michel Marcus, Sep 07 2018

Python

from sympy import divisors
def a(n): divs = divisors(n); s = sum(divs); return sum(s%d for d in divs)
print([a(n) for n in range(1, 73)]) # Michael S. Branicky, Nov 27 2021

Formula

a(A007691(n)) = 0.

a(A000040(n)) = 1.

a(A008578(n)) = tau(n) - 1.

a(n) = n for numbers 4, 45, 6048, 14421, ...