A318492 a(n) is the denominator of Sum_{d|n} Sum_{j|d} 1/j.

1, 2, 3, 4, 5, 6, 7, 8, 9, 2, 11, 12, 13, 14, 15, 16, 17, 9, 19, 20, 1, 22, 23, 24, 25, 26, 27, 28, 29, 6, 31, 32, 33, 34, 7, 18, 37, 38, 13, 40, 41, 2, 43, 44, 45, 46, 47, 16, 49, 5, 51, 52, 53, 27, 5, 8, 19, 58, 59, 60, 61, 62, 21, 64, 65, 66, 67, 4, 69, 14, 71, 36, 73, 74, 75

Offset

1,2

Links

Table of n, a(n) for n=1..75.

Formula

Denominators of coefficients in expansion of Sum_{k>=1} sigma(k)*x^k/(k*(1 - x^k)), where sigma(k) = sum of divisors of k (A000203).

Denominators of coefficients in expansion of -log(Product_{k>=1} (1 - x^k)^tau(k)), where tau(k) = number of divisors of k (A000005).

a(n) = denominator of Sum_{d|n} sigma(d)/d.

a(n) = denominator of (1/n)*Sum_{d|n} d*tau(d).

Example

1, 5/2, 7/3, 17/4, 11/5, 35/6, 15/7, 49/8, 34/9, 11/2, 23/11, 119/12, 27/13, 75/14, 77/15, 129/16, ...

Mathematica

Denominator[Table[Sum[DivisorSigma[-1, d], {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[DivisorSigma[1, d]/d, {d, Divisors[n]}], {n, 75}]]
Denominator[Table[Sum[d DivisorSigma[0, d], {d, Divisors[n]}]/n, {n, 75}]]
nmax = 75; Rest[Denominator[CoefficientList[Series[Sum[DivisorSigma[1, k] x^k/(k (1 - x^k)), {k, 1, nmax}], {x, 0, nmax}], x]]]
nmax = 75; Rest[Denominator[CoefficientList[Series[-Log[Product[(1 - x^k)^DivisorSigma[0, k], {k, 1, nmax}]], {x, 0, nmax}], x]]]

Prog

(PARI) a(n) = denominator(sumdiv(n, d, sumdiv(d, j, 1/j))); \\ Michel Marcus, Aug 28 2018

Crossrefs

Cf. A000005, A000203, A006171, A007429, A017665, A017666, A060640, A068986 (positions of 1's), A318491 (numerators).

Sequence in context: A356234 A284255 A284253 * A061499 A178051 A245346

Adjacent sequences: A318489 A318490 A318491 * A318493 A318494 A318495

Keyword

nonn,frac

Author

Ilya Gutkovskiy, Aug 27 2018

Status

approved