A212718 -id:A212718 - cp's OEIS frontend

A212717 Numerator of Sum_{k=1..n} 1/sigma(k).

1, 4, 19, 145, 53, 83, 353, 607, 8171, 75359, 78089, 79259, 11657, 2963, 12047, 378137, 386197, 389917, 397171, 2804377, 11344453, 11457293, 11626553, 11694257, 11825297, 11922017, 12023573, 12096113, 12231521, 12287941, 6207443, 6239683, 3140999, 9479417

Offset: 1

Michel Lagneau, May 25 2012

			1, 4/3, 19/12, 145/84, 53/28, 83/42, 353/168, ...

Harvey P. Dale, Table of n, a(n) for n = 1..1000
V. Sita Ramaiah and D. Suryanarayana, Sums of reciprocals of some multiplicative functions, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164.
László Tóth, Alternating Sums Concerning Multiplicative Arithmetic Functions, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1.

Cf. A000203, A212718 (denominators), A308039, A345327.

with(numtheory): a:=n->numer(sum(1/sigma(k), k=1..n)): seq(a(n), n=1..50);

Numerator[Table[Sum[1/DivisorSigma[1,k],{k,1,n}],{n,1,50}]]
Accumulate[1/DivisorSigma[1,Range[40]]]//Numerator (* Harvey P. Dale, Aug 13 2023 *)

a(n)/A212718(n) = c * (log(n) + gamma + Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p))) + O(log(n)^(2/3)*log(log(n))^(4/3)/n), where alpha(p) = 1 - ((p-1)^2/p) * Sum_{k>=1} 1/((p^k-1)*(p^(k+1)-1)), beta(p) = Sum_{k>=1} k/((p^k-1)*(p^(k+1)-1)), and c = Product_{p prime} alpha(p) = A308039 (Sita Ramaiah and Suryanarayana, 1979). - Amiram Eldar, Oct 16 2022

A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Original entry on oeis.org

1, 3, 12, 84, 84, 7, 56, 840, 10920, 32760, 32760, 32760, 32760, 16380, 32760, 1015560, 338520, 338520, 338520, 338520, 1354080, 4062240, 4062240, 4062240, 131040, 131040, 131040, 131040, 131040, 43680, 21840, 65520, 32760, 98280, 196560, 196560, 3734640, 3734640

Offset: 1

Amiram Eldar, Oct 16 2022

See A357845 for more details.

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. A000203, A068762, A357845 (numerators).

Similar sequence: A104529, A212718, A357821.

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

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

from fractions import Fraction
from sympy import divisor_sigma
def A357846(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(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)/sigma(k)), where sigma(k) = A000203(k).

A067577 a(n) = Product_{i=1..n} sigma(i) * Sum_{i=1..n} 1/sigma(i).

Original entry on oeis.org

1, 4, 19, 145, 954, 11952, 101664, 1573344, 21179232, 390661056, 4857760512, 138055228416, 1989835352064, 48554918608896, 1184490930438144, 37179368055373824, 683493250562260992, 26913000032107757568, 548273767789158727680, 23227773590084738088960, 751700319286194622955520

Offset: 1

Benoit Cloitre, Jan 30 2002

nonn

Michael De Vlieger, Table of n, a(n) for n = 1..419

Cf. A000203 (sigma), A066780, A212717, A212718.

Array[Product[DivisorSigma[1, i], {i, #}]*Sum[1/DivisorSigma[1, j], {j, #}] &, 21] (* Michael De Vlieger, Jul 16 2022 *)

a(n) = prod(i=1, n, sigma(i)) * sum(i=1, n, 1/sigma(i)); \\ Michel Marcus, Jan 09 2021

More terms from Michel Marcus, Jan 09 2021

A357819 Denominators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 3, 12, 4, 12, 1, 8, 24, 24, 72, 72, 36, 252, 504, 126, 504, 504, 504, 2520, 840, 3360, 10080, 10080, 10080, 2016, 2016, 2016, 2016, 10080, 10080, 5040, 1260, 5040, 15120, 7560, 3780, 71820, 17955, 143640, 17955, 35910, 574560, 6320160, 6320160, 6320160, 6320160

Offset: 1

Amiram Eldar, Oct 14 2022

See A357818 for more details.

Cf. A001615, A173290, A357818 (numerators).

Similar sequences: A048049, A104529, A212718.

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

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

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

cp's OEIS Frontend

A212717 Numerator of Sum_{k=1..n} 1/sigma(k).

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A357846 Denominators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

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A067577 a(n) = Product_{i=1..n} sigma(i) * Sum_{i=1..n} 1/sigma(i).

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A357819 Denominators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

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