A212717 -id:A212717 - cp's OEIS frontend

A212718 Denominator of Sum_{k=1..n} 1/sigma(k).

1, 3, 12, 84, 28, 42, 168, 280, 3640, 32760, 32760, 32760, 4680, 1170, 4680, 145080, 145080, 145080, 145080, 1015560, 4062240, 4062240, 4062240, 4062240, 4062240, 4062240, 4062240, 4062240, 4062240, 4062240, 2031120, 2031120, 1015560, 3046680, 6093360, 6093360

Offset: 1

Michel Lagneau, May 25 2012

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

Cf. A000203, A212717.

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

Denominator[Table[Sum[1/DivisorSigma[1,k],{k,1,n}],{n,1,50}]]

A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

Original entry on oeis.org

1, 7, 29, 129, 571, 2525, 11167, 49372, 218295, 965177, 4267457, 18868240, 83424514, 368855252, 1630865929, 7210751807, 31881800153

Offset: 1

Labos Elemer, Aug 29 2002

Jean-Marie De Koninck, Ces nombres qui nous fascinent, Entry 129, p. 44, Ellipses, Paris, 2008.

Cf. A000203 (sigma), A002387, A004080, A046024, A074631, A074633, A074467, A212717, A308039.

{s=0, s1=0}; Do[s=s+(1/DivisorSigma[1, n]); If[Greater[Floor[s], s1], s1=Floor[s]; Print[{n, Floor[s]}]], {n, 1, 1000000}]

Limit_{n->oo} a(n+1)/a(n) = exp(1/c) = 4.42142525588146107878... where c = A308039. - Amiram Eldar, May 05 2024

2 more terms from Lekraj Beedassy, Jul 14 2008
a(11)-a(15) from Donovan Johnson, Aug 22 2011
a(16)-a(17) from Amiram Eldar, May 05 2024

A357845 Numerators of the partial alternating sums of the reciprocals of the sum of divisors function (A000203).

Original entry on oeis.org

1, 2, 11, 65, 79, 6, 55, 769, 10837, 30691, 33421, 32251, 34591, 16613, 34591, 1039561, 365327, 356647, 373573, 365513, 1504367, 4400261, 4569521, 4501817, 149447, 146327, 149603, 147263, 151631, 49937, 25651, 75913, 38639, 114097, 232289, 230129, 4470731, 4408487

Offset: 1

Amiram Eldar, Oct 16 2022

			Fractions begin with 1, 2/3, 11/12, 65/84, 79/84, 6/7, 55/56, 769/840, 10837/10920, 30691/32760, 33421/32760, 32251/32760, ...

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, A065442, A065443, A068762, A357846 (denominators).

Similar sequence: A104528, A212717, A357820.

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

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

from fractions import Fraction
from sympy import divisor_sigma
def A357845(n): return sum(Fraction(1 if k&1 else -1, divisor_sigma(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)/sigma(k)), where sigma(k) = A000203(k).

a(n)/A357846(n) ~ E * ((2/K-1)*(log(n) + gamma + F) + 2*log(2)*K'/K^2) + O(log(n)^(5/3)*log(log(n))^(4/3)/n), where E = Product_{p prime} alpha(p), F = Sum_{p prime} (p-1)^2*beta(p)*log(p)/(p*alpha(p)), 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)), K = A065442, K' = A065443 (Tóth, 2017).

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

A357818 Numerators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

Original entry on oeis.org

1, 4, 19, 7, 23, 2, 17, 53, 55, 169, 175, 89, 641, 1303, 331, 1345, 1373, 1387, 7061, 2377, 9613, 29119, 29539, 29749, 6017, 6065, 6121, 6163, 31151, 31291, 15803, 3977, 16013, 48319, 24317, 12211, 233899, 58774, 472757, 59344, 119543, 1918673, 21249043, 21336823

Offset: 1

Amiram Eldar, Oct 14 2022

			Fractions begin with 1, 4/3, 19/12, 7/4, 23/12, 2, 17/8, 53/24, 55/24, 169/72, 175/72, 89/36, ...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 100, p. 169.
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. A001615, A173290, A357819 (denominators).
Cf. A001620, A065463, A335707.
Similar sequences: A028415, A104528, A212717.

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

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

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

a(n)/A357819(n) ~ C * (log(n) + gamma + D) + 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) (Sita Ramaiah and Suryanarayana, 1979; Tóth, 2017).

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A212718 Denominator of Sum_{k=1..n} 1/sigma(k).

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A074468 Least number m such that the Sigma-Harmonic sequence Sum_{k=1..m} 1/sigma(k) >= n.

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A357845 Numerators 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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A357818 Numerators of the partial sums of the reciprocals of the Dedekind psi function (A001615).

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