A001008 -id:A001008 - cp's OEIS frontend

A103930 Numerators of squares of harmonic numbers A001008/A002805.

1, 9, 121, 625, 18769, 2401, 131769, 579121, 50822641, 54479161, 7007531521, 7399612441, 1313299956049, 1372958223289, 1429834803049, 5936819760481, 1775966959381729, 203755669038601, 75787776947048401, 3117562300468225

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding denominators are given in A103931.

W. Lang: Rationals H(n)^2.

A103930 := n->numer(sum(1/i, i=1..n)^2); seq(A103930(k), k=1..40); # Wesley Ivan Hurt, Sep 30 2013

a[n_] := HarmonicNumber[n]^2 // Numerator; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Sep 16 2013 *)

G.f.: -((d^3/dx^3)((log(1-x))^3))/3 + dilog(1-x)/(1-x) = ((log(1-x)^2) + dilog(1-x))/(1-x) with dilog(1-x)=polylog(2, x).
First differences give A103932(n)/A103933(n).
a(n) = numerator(H(n)^2), with the harmonic numbers H(n) = A001008(n)/A002805(n), n >= 1.

A349851 Decimal expansion of Sum_{k>=1} H(k)*L(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and L(k) = A000032(k) is the k-th Lucas number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Dec 02 2021

			8.46297249299971224539772505825511366269870763156442...

Hideyuki Ohtsuka, Problem B-1200, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 54, No. 4 (2016), p. 367; Harmonic and Fiboancci [sic]/Lucas Numbers, Solution to Problem B-1200 by Kenny B. Davenport, ibid., Vol. 55, No. 4 (2017), pp. 372-373.

Cf. A000032, A001008, A001622, A002162, A002390, A002805, A349850.

RealDigits[6*Log[2] + 4*Sqrt[5]*Log[GoldenRatio], 10, 100][[1]]

Equals log(64*phi^(4*sqrt(5))) = 6*log(2) + 4*sqrt(5)*log(phi), where phi is the golden ratio (A001622).

A035047 Denominators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

Original entry on oeis.org

1, 2, 3, 4, 15, 12, 105, 24, 315, 120, 3465, 40, 45045, 280, 45045, 560, 765765, 5040, 14549535, 5040, 14549535, 55440, 334639305, 55440, 1673196525, 720720, 5019589575, 720720, 145568097675, 720720

Offset: 1

N. J. A. Sloane

Robert Israel, Table of n, a(n) for n = 1..2000
N. J. A. Sloane, Transforms

Cf. A035048.

S:= series(log(1-x)/(x^2-1), x, 101):
seq(denom(coeff(S,x,j)),j=1..100); # Robert Israel, Jun 02 2015

a(n)=denominator(polcoeff(log(1-x)/(x^2-1)+O(x^(n+1)),n))

G.f. for A035048(n)/A035047(n) : log(1-x)/(x^2-1). - Benoit Cloitre, Jun 15 2003

a(n) = denominator((-1)^(n+1)*1/2*(log(2)+(-1)^(n+1)*(gamma+1/2*(psi(1+n/2)-psi(3/2+n/2))+psi(2+n)))), with gamma the Euler-Mascheroni constant. - [Gerry Martens, Apr 28 2011]

A065454 Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).

Original entry on oeis.org

9, 11, 13, 14, 21, 25, 27, 29, 33, 34, 35, 37, 38, 39, 44, 45, 47, 49, 50, 51, 54, 55, 56, 57, 59, 61, 64, 67, 69, 73, 74, 75, 77, 79, 81, 83, 84, 85, 86, 89, 90, 91, 92, 93, 94, 95, 97, 98, 101, 103, 105, 107, 110, 111, 113, 114, 115, 116, 117, 118, 121, 122, 123, 125

Offset: 1

Benoit Cloitre, Nov 24 2001

nonn

Shiu (2016) proved that this sequence is infinite. Wu and Chen (2019) proved that the asymptotic density of this sequence is 1. - Amiram Eldar, Jan 29 2021

			For example: H(11) = 83711/27720, H(12) = 86021/27720 and so a(2) = 11.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Shiu, The denominators of harmonic numbers, arXiv:1607.02863 [math.NT], 2016.
Bing-Ling Wu and Yong-Gao Chen, On the denominators of harmonic numbers, II, Journal of Number Theory, Vol. 200 (2019), pp. 397-406.

Cf. A001008, A002805.

Position[Partition[Denominator @ HarmonicNumber[Range[126]], 2, 1], {x_, x_}] // Flatten (* Amiram Eldar, Jan 29 2021 *)

A093569 For p = prime(n), the number of integers k < p-1 such that p divides A001008(k), the numerator of the harmonic number H(k).

Original entry on oeis.org

0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 6, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0

Offset: 1

T. D. Noe, Apr 01 2004

nonn

It is well-known that prime p >= 3 divides the numerator of H(p-1). For primes p in A092194, there are integers k < p-1 for which p divides the numerator of H(k). Interestingly, if p divides A001008(k) for k < p-1, then p divides A001008(p-k-1). Hence the terms of this sequence are usually even. The only exceptions are the two known Wieferich primes 1093 and 3511, A001220, which have 3 values of k < p-1 for which p divides A001008(k), one being k = (p-1)/2.

			a(5) = 2 because 11 = prime(5) and there are 2 values, k = 3 and 7, such that 11 divides A001008(k).

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Wieferich Prime

Cf. A001008, A001220, A092194.

len=500; Table[p=Prime[i]; cnt=0; k=1; While[k

A103931 Denominators of squares of harmonic numbers A001008/A002805.

Original entry on oeis.org

1, 4, 36, 144, 3600, 400, 19600, 78400, 6350400, 6350400, 768398400, 768398400, 129859329600, 129859329600, 129859329600, 519437318400, 150117385017600, 16679709446400, 6021375110150400, 240855004406016, 26761667156224

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding numerators are given in A103930. For the rational see the link there.

Denominator[HarmonicNumber[Range[30]]^2] (* Harvey P. Dale, Oct 16 2022 *)

a(n)=denominator(H(n)^2), with the harmonic numbers H(n)=A001008(n)/A002805(n), n>=1.

A103933 Denominators of first difference of squares of harmonic numbers A001008/A002805.

Original entry on oeis.org

1, 4, 9, 48, 150, 90, 490, 2240, 11340, 2520, 152460, 83160, 2342340, 2522520, 540540, 11531520, 104144040, 110270160, 737176440, 775975200, 162954792, 56904848, 1368302936, 2141691552, 111546435000, 116008292400, 1084231348200

Offset: 1

Wolfdieter Lang, Mar 24 2005

The corresponding numerators are given in A103932. For the rationals see the link there.

Denominator[Differences[HarmonicNumber[Range[0,30]]^2]] (* Harvey P. Dale, Sep 09 2012 *)

a(n)=numerator(r(n)), with the rationals r(n)=H(n)^2-H(n-1)^2 where H(n)= A001008(n)/A002805(n), n>=1, H(0):=0.

Offset corrected by Mohammed Yaseen, Aug 09 2023

A331777 Numerators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

Original entry on oeis.org

1, 1, 1, 0, -1, 1, -1, -43, 1831, 949, -137309, -85511, 3404045159, 777985057, -21024051077, -2192231411, 467347169033357, 10187765700589, -11741590582705819219, -3086703970985605357, 169597995722575162268081, 19606186988235984155519, -62715098968866173387571821

Offset: 0

N. J. A. Sloane, Feb 09 2020

Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, 40:2 (2016), 279-290. See Eq. (2.11).

Denominators are in A331778.

Cf. A001008, A001620, A002805, A093334, A238813, A331943.

Numerator[CoefficientList[Series[Exp[2*(HarmonicNumber[k] - EulerGamma)]/k^2, {k, Infinity, 25}], 1/k]] (* Vaclav Kotesovec, Feb 10 2020 *)

Sign of a(7) corrected and more terms from Vaclav Kotesovec, Feb 10 2020

A331778 Denominators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

Original entry on oeis.org

1, 1, 3, 1, 90, 90, 567, 5670, 340200, 113400, 11226600, 5613300, 91945854000, 18389170800, 137918781000, 13135122000, 562708626480000, 11483849520000, 2020686677689680000, 505171669422420000, 3334133018187972000000, 370459224243108000000, 115027589127485034000000

Offset: 0

N. J. A. Sloane, Feb 09 2020

Chao-Ping Chen, On the coefficients of asymptotic expansion for the harmonic number by Ramanujan, The Ramanujan Journal, 40:2 (2016), 279-290. See Eq. (2.11).

Numerators are in A331777.

Cf. A001008, A001620, A002805, A093334, A238813.

Denominator[CoefficientList[Series[Exp[2*(HarmonicNumber[k] - EulerGamma)]/k^2, {k, Infinity, 25}], 1/k]] (* Vaclav Kotesovec, Feb 10 2020 *)

More terms from Vaclav Kotesovec, Feb 10 2020

A363538 Decimal expansion of Sum_{k>=1} (H(k) - log(k) - gamma)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Jun 09 2023

			0.72869391700393060593760589102029180041750271881292...

Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in Pi^(-2) and into the formal enveloping series with rational coefficients only, Journal of Number Theory, Vol. 158 (2016), pp. 365-396.
Ovidiu Furdui, Problem 844, Problems and Solutions, The College Mathematics Journal, Vol. 38, No. 1 (2007), p. 61; Infinite sums and Euler's constant, Solution to Problem 844, ibid., Vol. 39, No. 1 (2008), pp. 71-72.

Cf. A001008, A001620, A002805, A072691, A082633, A363539, A363540.

RealDigits[-StieltjesGamma[1] - EulerGamma^2/2 + Pi^2/12, 10, 120][[1]]

Equals -gamma_1 - gamma^2/2 + Pi^2/12, where gamma_1 is the 1st Stieltjes constant (A082633).

cp's OEIS Frontend

A103930 Numerators of squares of harmonic numbers A001008/A002805.

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A349851 Decimal expansion of Sum_{k>=1} H(k)*L(k)/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and L(k) = A000032(k) is the k-th Lucas number.

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A035047 Denominators of alternating sum transform (PSumSIGN) of Harmonic numbers H(n) = A001008/A002805.

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A065454 Let the k-th harmonic number be H(k) = Sum_{i=1..k} 1/i = P(k)/Q(k) = A001008(k)/A002805(k); sequence gives values of k such that Q(k) = Q(k+1).

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A093569 For p = prime(n), the number of integers k < p-1 such that p divides A001008(k), the numerator of the harmonic number H(k).

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A103931 Denominators of squares of harmonic numbers A001008/A002805.

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A103933 Denominators of first difference of squares of harmonic numbers A001008/A002805.

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A331777 Numerators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

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A331778 Denominators of coefficients in asymptotic expansion of exp(2*(H_k-gamma))/k^2 in powers of 1/k, where H_k are the harmonic numbers A001008/A002805 and gamma is the Euler-Mascheroni constant A001620.

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