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

1,4

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)

Links

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

Formula

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

Example

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

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 *)

Prog

(Magma) [(&+[SumOfDivisors(d) mod d: d in Divisors(n)]): n in [1..100]]
(PARI) a(n) = sumdiv(n, d, sigma(d) % d); \\ Michel Marcus, Mar 11 2018

Crossrefs

Cf. A008578, A054024.

Sequence in context: A337515 A090885 A008476 * A370120 A112621 A081448

Adjacent sequences: A300654 A300655 A300656 * A300658 A300659 A300660

Keyword

nonn

Author

Jaroslav Krizek, Mar 10 2018

Status

approved