A064609 -id:A064609 - cp's OEIS frontend

A379516 Denominators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

1, 3, 12, 60, 60, 5, 40, 360, 360, 120, 120, 120, 840, 140, 840, 14280, 42840, 42840, 42840, 8568, 34272, 34272, 34272, 11424, 148512, 49504, 7072, 35360, 106080, 318240, 159120, 1750320, 109395, 656370, 5250960, 26254800, 498841200, 498841200, 3491888400, 3491888400

Offset: 1

Amiram Eldar, Dec 23 2024

Amiram Eldar, Table of n, a(n) for n = 1..1000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, Journal of Integer Sequences, Vol. 16 (2013), Article 13.6.3. See p. 4, eq. (vi).
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1. See section 4.9, pp. 28-29.

Cf. A034448, A064609, A370898, A379514, A379515 (numerators).

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; Denominator[Accumulate[Table[(-1)^(n+1)/usigma[n], {n, 1, 50}]]]

usigma(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + f[i, 1]^f[i, 2]);}
list(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / usigma(k); print1(denominator(s), ", "))};

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/A034448(k)).

A335005 Decimal expansion of Pi^2/(12*zeta(3)).

Original entry on oeis.org

6, 8, 4, 2, 1, 6, 3, 8, 8, 8, 1, 0, 1, 0, 2, 9, 3, 7, 8, 6, 8, 3, 8, 2, 9, 2, 6, 9, 9, 2, 3, 9, 5, 9, 7, 0, 5, 6, 5, 4, 0, 6, 9, 5, 7, 3, 2, 6, 2, 0, 6, 9, 6, 1, 0, 3, 8, 6, 7, 6, 5, 9, 6, 3, 8, 4, 1, 7, 2, 4, 8, 9, 8, 9, 3, 8, 0, 0, 9, 7, 1, 1, 4, 1, 1, 0, 1

Offset: 0

Amiram Eldar, May 19 2020

			0.68421638881010293786838292699239597056540695732620...

Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
R. Sitaramachandrarao and D. Suryanarayana, On Sigma_{n<=x} sigma*(n) and Sigma_{n<=x} phi*(n), Proceedings of the American Mathematical Society, Vol. 41, No. 1 (1973), pp. 61-66.

Cf. A002117(zeta(3)), A013661 (zeta(2)), A034448, A064609, A072691 (Pi^2/12), A253905 (zeta(3)/zeta(2)).

RealDigits[Pi^2/12/Zeta[3], 10, 100][[1]]

Pi^2/(12*zeta(3)) \\ Michel Marcus, May 19 2020

Equals lim_{k->oo} A064609(k)/k^2, where A064609(k) is the partial sums of A034448, the sum of unitary divisors from 1 to k.

Equals zeta(2)/(2*zeta(3)) = A013661/(2*A002117) = A072691/A002117 = 1/(2*A253905).

cp's OEIS Frontend

A379516 Denominators of the partial alternating sums of the reciprocals of the sum of unitary divisors function (A034448).

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