A211177 -id:A211177 - cp's OEIS frontend

A211178 Denominator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

1, 1, 2, 1, 4, 4, 12, 3, 6, 12, 60, 30, 60, 20, 40, 20, 80, 240, 720, 720, 720, 144, 1584, 1584, 7920, 7920, 7920, 7920, 55440, 55440, 11088, 5544, 27720, 55440, 55440, 55440, 55440, 6160, 18480, 2310, 9240, 9240, 3080, 3080, 1155, 210, 2415, 38640, 5520, 5520

Offset: 1

Benoit Cloitre, Feb 01 2013

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.

Cf. A000010, A082695, A211177 (numerators).

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

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

A211177(n)/a(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 terms from Amiram Eldar, Nov 20 2020

A357820 Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 2, 11, 3, 11, 5, 23, 7, 23, 65, 71, 17, 64, 491, 64, 491, 173, 505, 2651, 2581, 10639, 1151, 3593, 3523, 727, 237, 2189, 2147, 11071, 10931, 5623, 2759, 5623, 16589, 2113, 8347, 162373, 159979, 20318, 160549, 163969, 649891, 7292441, 7204661, 7292441, 7204661

Offset: 1

Amiram Eldar, Oct 14 2022

			Fractions begin with 1, 2/3, 11/12, 3/4, 11/12, 5/6, 23/24, 7/8, 23/24, 65/72, 71/72, 17/18, ...

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

Cf. A001615, A173290, A357821 (denominators).
Cf. A001620, A065463, A335707.
Similar sequence: A211177.

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

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

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

a(n)/A357821(n) ~ (C/5) * (log(n) + gamma + D + 24*log(2)/5) + O(log(n)^(2/3) * log(log(n))^(4/3) / n), where C = Product_{p prime} (1 - 1/(p*(p+1))) (A065463), and D = Sum_{p prime} log(p)/(p^2+p-1) (A335707) (Bordellès and Cloitre, 2013; Tóth, 2017).

A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

Original entry on oeis.org

1, 1, 1, 2, 7, 11, 17, 7, 3, 5, 7, 19, 25, 11, 25, 113, 143, 133, 163, 51, 14, 51, 61, 117, 391, 361, 391, 371, 431, 52, 119, 19, 81, 19, 81, 709, 799, 377, 799, 1553, 1733, 211, 467, 226, 467, 889, 979, 961, 1021, 991, 259, 503, 274, 2147, 2237, 274, 1141, 274

Offset: 1

Amiram Eldar, Oct 16 2022

			Fractions begin with 1, 1/2, 1, 2/3, 7/6, 11/12, 17/12, 7/6, 3/2, 5/4, 7/4, 19/12, ...

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

Cf. A000005, A307704, A357844 (denominators).

Similar sequences: A104528, A211177, A357820.

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

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

from fractions import Fraction
from sympy import divisor_count
def A357843(n): return sum(Fraction(1 if k&1 else -1, divisor_count(k)) for k in range(1,n+1)).numerator # Chai Wah Wu, Oct 16 2022

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

a(n)/A357844(n) ~ n * Sum_{k=1..N} B_k/log(n)^(k-1/2) + O(n/log(n)^(N+1/2)), where B_k are constants, and in particular B_1 = (1/log(2) - 1) * (1/sqrt(Pi)) * Product_{p prime} sqrt(p^2-p) * log(p/(p-1)) (Tóth, 2017).

cp's OEIS Frontend

A211178 Denominator of Sum_{k=1..n}(-1)^k/phi(k), where phi = A000010.

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A357820 Numerators of the partial alternating sums of the reciprocals of the Dedekind psi function (A001615).

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A357843 Numerators of the partial alternating sums of the reciprocals of the number of divisors function (A000005).

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