A211178 -id:A211178 - cp's OEIS frontend

A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

-1, 0, -1, 0, -1, 1, 1, 1, 1, 5, 19, 17, 29, 13, 21, 13, 47, 181, 503, 593, 533, 121, 1259, 1457, 6889, 7549, 7109, 7769, 52403, 59333, 11497, 6095, 29089, 61643, 59333, 63953, 62413, 7277, 21061, 2777, 10877, 11647, 3809, 3963, 1438, 271, 3064, 51439, 7217, 7493

Offset: 1

Benoit Cloitre, Feb 01 2013

			Fractions begin with -1, 0, -1/2, 0, -1/4, 1/4, 1/12, 1/3, 1/6, 5/12, 19/60, 17/30, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Olivier Bordellès and Benoit Cloitre, An Alternating Sum Involving the Reciprocal of Certain Multiplicative Functions, J. Int. Seq., Vol. 16 (2013) Article 13.6.3.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000010, A028415, A211178 (denominators).

Cf. A001620, A082695, A085609.

Numerator @ Accumulate[Table[(-1)^k/EulerPhi[k], {k, 1, 50}]] (* Amiram Eldar, Nov 20 2020 *)

a(n)=numerator(sum(k=1,n,(-1)^k/eulerphi(k)))

a(n)/A211178(n) = c*log(n) + O(1) with a suitable constant c (see ref).
The constant above is c = zeta(2)*zeta(3)/(3*zeta(6)) = (1/3) * A082695. - Amiram Eldar, Nov 20 2020
More accurately, a(n)/A211178(n) ~ (A/3) * (log(n) + gamma - B - 8*log(2)/3) + O(log(n)^(5/3)/n), where A = zeta(2)*zeta(3)/zeta(6) (A082695), gamma is Euler's constant (A001620), and B = Sum_{p prime} log(p)/(p^2-p+1) (A085609) (Bordellès and Cloitre, 2013; Tóth, 2017). - Amiram Eldar, Oct 14 2022

More terms from Amiram Eldar, Nov 20 2020

A357821 Denominators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 4, 12, 6, 24, 8, 24, 72, 72, 18, 63, 504, 63, 504, 168, 504, 2520, 2520, 10080, 1120, 3360, 3360, 672, 224, 2016, 2016, 10080, 10080, 5040, 2520, 5040, 15120, 1890, 7560, 143640, 143640, 17955, 143640, 143640, 574560, 6320160, 6320160, 6320160, 6320160

Offset: 1

Amiram Eldar, Oct 14 2022

See A357820 for more details.

Cf. A001615, A173290, A357820 (numerators).

Similar sequence: A211178.

psi[n_] := n * Times @@ (1 + 1/Transpose[FactorInteger[n]][[1]]); psi[1] = 1; Denominator[Accumulate[1/Array[(-1)^(# + 1)*psi[#] &, 50]]]

f(n) = n * sumdivmult(n, d, issquarefree(d)/d); \\ A001615
a(n) = denominator(sum(k=1, n, (-1)^(k+1)/f(k))); \\ Michel Marcus, Oct 15 2022

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

A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Original entry on oeis.org

1, 2, 1, 3, 6, 12, 12, 6, 2, 4, 4, 12, 12, 6, 12, 60, 60, 60, 60, 20, 5, 20, 20, 40, 120, 120, 120, 120, 120, 15, 30, 5, 20, 5, 20, 180, 180, 90, 180, 360, 360, 45, 90, 45, 90, 180, 180, 180, 180, 180, 45, 90, 45, 360, 360, 45, 180, 45, 90, 180, 180, 45, 90, 630

Offset: 1

Amiram Eldar, Oct 16 2022

See A357843 for more details.

László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000005, A307704, A357843 (numerators).

Similar sequences: A104529, A211178, A357821.

Denominator[Accumulate[Array[(-1)^(# + 1)/DivisorSigma[0, #] &, 60]]]

lista(nmax) = {my(s = 0); for(k = 1, nmax, s += (-1)^(k+1) / numdiv(k); print1(denominator(s), ", "))};

from fractions import Fraction
from sympy import divisor_count
def A357844(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1,n+1)).denominator # Chai Wah Wu, Oct 16 2022

a(n) = denominator(Sum_{k=1..n} (-1)^(k+1)/d(k)), where d(k) = A000005(k).

cp's OEIS Frontend

A211177 Numerator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

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A357821 Denominators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

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Author

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Programs

Mathematica

PARI

Formula

A357844 Denominators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

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Author

Keywords

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Programs

Mathematica

PARI

Python

Formula