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A300657 a(n) = Sum_{d|n} sigma(d) mod d. 2
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 (list; graph; refs; listen; history; text; internal format)
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 * A112621 A081448 A322906

Adjacent sequences:  A300654 A300655 A300656 * A300658 A300659 A300660

KEYWORD

nonn

AUTHOR

Jaroslav Krizek, Mar 10 2018

STATUS

approved

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Last modified August 4 21:18 EDT 2021. Contains 346455 sequences. (Running on oeis4.)