A002805 -id:A002805 - cp's OEIS frontend

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.

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).

A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/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

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

Offset: 1

Amiram Eldar, Jun 09 2023

The formula for this sum was found by Olivier Oloa and proved by Roberto Tauraso in 2014.

			1.96896908391052854646489145379668054231137794286819...

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.
Olivier Oloa, A closed form of the series Sum_{n=1..oo} (H(n)^2 - (gamma + ln(n))^2)/n, Mathematics StackExchange, 2014.

Cf. A001008, A001620, A002117, A002805, A082633, A086279, A363538, A363540.

RealDigits[-StieltjesGamma[2] - 2*EulerGamma*StieltjesGamma[1] - 2*EulerGamma^3/3 + 5*Zeta[3]/3, 10, 120][[1]]

Equals -gamma_2 - 2*gamma*gamma_1 - (2/3)*gamma^3 + (5/3)*zeta(3), where gamma_1 and gamma_2 are the 1st and 2nd Stieltjes constants (A082633, A086279).

A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/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

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

Offset: 1

Amiram Eldar, Jun 09 2023

			5.82174008504864652889686861550204134315033324319577...

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.

Cf. A001008, A001620, A002117, A013662, A082633, A086279, A086280, A363538, A363539.

RealDigits[-StieltjesGamma[3] - 3*EulerGamma*StieltjesGamma[2] - 3*EulerGamma^2*StieltjesGamma[1] - 3*EulerGamma^4/4 + 43*Zeta[4]/8, 10, 120][[1]]

Equals -gamma_3 - 3*gamma*gamma_2 - 3*gamma^2*gamma_1 - (3/4)*gamma^4 + (43/8)*zeta(4), where gamma_1, gamma_2 and gamma_3 are the 1st, 2nd and 3rd Stieltjes constants (A082633, A086279, A086280).

A120308 Numerator((p-1)*H(p-1))/p^2 for p = prime(n) > 3, where H(k) is k-th harmonic number A001008(k)/A002805(k).

Original entry on oeis.org

1, 3, 61, 509, 8431, 118623, 36093, 375035183, 9682292227, 40030624861, 1236275063173, 46600968591317, 2690511212793403, 5006621632408586951, 73077117446662772669, 4062642402613316532391, 139715526178793824689891

Offset: 3

Alexander Adamchuk, Jul 16 2006

G. C. Greubel, Table of n, a(n) for n = 3..344

Cf. A096617, A001008, A002805.

[Numerator((NthPrime(n)-1)*HarmonicNumber(NthPrime(n)-1)/NthPrime(n)^2): n in [3..25]]; // G. C. Greubel, Sep 02 2018

N:= 50: # to get the first N terms
Primes:= select(isprime,[seq(2*i+1,i=2..(ithprime(N+2)-1)/2)]):
H:= ListTools[PartialSums]([seq(1/i,i=1..Primes[-1]-1)]):
seq(numer((p-1)*H[p-1])/p^2, p=Primes); # Robert Israel, Sep 09 2014

Numerator[Table[(Prime[n]-1)*(Sum[(1/k), {k, 1, Prime[n]-1}]),{n,3,20}]]/Table[Prime[n]^2,{n,3,20}]
Table[((p-1)HarmonicNumber[p-1])/p^2,{p,Prime[Range[2,20]]}]//Numerator (* Harvey P. Dale, May 19 2021 *)

{a(n) = numerator((prime(n)-1)*sum(k=1,prime(n)-1, 1/k)/prime(n)^2)};
for(n=3,25, print1(a(n), ", ")) \\ G. C. Greubel, Sep 02 2018

a(n) = numerator((prime(n)-1)*(Sum_{k=1..prime(n)-1} 1/k))/prime(n)^2 for n > 2.

a(n) = A096617(p-1)/p^2 for p = prime(n) > 3.

A348373 Decimal expansion of Sum_{k>=1} H(k)^2/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Oct 15 2021

			2.12538708076642786113951769297269016094950285280134...

István Mező, A q-Raabe formula and an integral of the fourth Jacobi theta function, Journal of Number Theory, Vol. 133, No. 2 (2013), pp. 692-704.
Seán Mark Stewart, Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbers, Tatra Mountains Mathematical Publications, Vol. 77, No. 1 (2020), pp. 73-98.

Cf. A001008, A002805, A013661, A103930, A103931, A253191.

Similar constants: A016627, A076788.

RealDigits[Pi^2/6 + Log[2]^2, 10, 100][[1]]

Equals Pi^2/6 + log(2)^2 = A013661 + A253191.

A360029 Consider a ruler composed of n segments with lengths 1, 1/2, 1/3, ..., 1/n with total length A001008(n)/A002805(n). a(n) is the minimum number of distinct distances of all pairs of marks that can be achieved by permuting the positions of the segments.

Original entry on oeis.org

1, 3, 6, 10, 15, 18, 25, 33, 42, 52, 63, 71, 84, 98, 107, 123, 140, 152, 171, 185, 198, 220, 243, 256, 281, 307, 334, 354, 383, 403, 434, 466, 489, 523, 552, 581, 618, 656, 695, 728

Offset: 1

Hugo Pfoertner, Jan 22 2023

Without permutation of the arrangement of the segments, the number of distinct distances between any pair of marks is n*(n+1)/2.

			a(6) = 18: permuted segment lengths 1, 1/4, 1/2, 1/3, 1/6, 1/5 -> marks at 0, 1, 5/4, 7/4, 25/12, 9/4, 49/20 -> 18 distinct distances 1/6, 1/5, 1/4, 1/3, 11/30, 1/2, 7/10, 3/4, 5/6, 1, 13/12, 6/5, 5/4, 29/20, 7/4, 25/12, 9/4, 49/20, whereas the non-permuted ruler with marks at 0, 1, 3/2, 11/6, 25/12, 137/60, 49/20 gives 21 distinct distances 1/6, 1/5, 1/4, 1/3, 11/30, 9/20, 1/2, 7/12, 37/60, 47/60, 5/6, 19/20, 1, 13/12, 77/60, 29/20, 3/2, 11/6, 25/12, 137/60, 49/20.

Hugo Pfoertner, Examples of rulers with the minimum number of measurable distances up to n=38, Feb 01 2023.

Cf. A000217, A001008, A002805, A003022.

a360029(n) = {if (n<=1, 1, my (mi=oo); w = vectorsmall(n-1, i, i+1);
forperm (w, p, my(v=vector(n,i,1/i), L=List(v)); for (m=1, n, v[m] = 1 + sum (k=1, m-1, 1/p[k]); listput(L, v[m])); for (i=1, n-1, for (j=i+1, n, listput (L, v[j]-v[i]))); mi = min(mi, #Set(L))); mi)};

a(39)-a(40) from Hugo Pfoertner, Feb 19 2023

A370742 Decimal expansion of Sum_{k>=2} H(k-1) * F(k) / (k*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

Original entry on oeis.org

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

Offset: 0

Amiram Eldar, Feb 29 2024

			0.59667348783398269737770682436833083924687967042183...

Kenny B. Davenport, Problem B-1222, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 1 (2018), p. 81; The Generating Function for Harmonic Numbers, Solution to Problem B-1222 by Amanda M. Andrews and Samantha L. Zimmerman, ibid., Vol. 57, No. 1 (2019), pp. 83-84.

Cf. A000045, A001008, A001622, A002162, A002163, A002390, A002805, A349850, A370743.

RealDigits[4 * Log[2] * Log[GoldenRatio] / Sqrt[5], 10, 120][[1]]

PARI
```
4 * log(2) * log(quadgen(5)) / sqrt(5)
```

Equals 4 * log(2) * log(phi) / sqrt(5), where phi is the golden ratio (A001622) (Davenport, 2018).

A370743 Decimal expansion of Sum_{k>=2} H(k-1) * L(k) / (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

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

Offset: 1

Amiram Eldar, Feb 29 2024

			1.40671229622697899465481881125279601179617835179174...

Kenny B. Davenport, Problem B-1222, Elementary Problems and Solutions, The Fibonacci Quarterly, Vol. 56, No. 1 (2018), p. 81; The Generating Function for Harmonic Numbers, Solution to Problem B-1222 by Amanda M. Andrews and Samantha L. Zimmerman, ibid., Vol. 57, No. 1 (2019), pp. 83-84.

Cf. A000032, A001008, A001622, A002162, A002390, A002805, A349851, A370742.

RealDigits[Log[2]^2 + 4*Log[GoldenRatio]^2, 10, 120][[1]]

PARI
```
log(2)^2 + 4*log(quadgen(5))^2
```

Equals log(2)^2 + 4*log(phi)^2, where phi is the golden ratio (A001622) (Davenport, 2018).

A248979 Numbers n such that 11 is not a divisor of A002805(11*n).

Original entry on oeis.org

0, 33, 77, 110, 847, 880, 924, 957, 1210, 1243, 1287, 1320, 9328, 9372, 9416, 9702, 9768, 10538, 10582, 10626, 14201, 14223, 102608, 102641, 102685, 102718, 103136, 103158, 116413, 116457, 116501, 156255, 156277, 1128688, 1128721, 1128765, 1128798, 1129073

Offset: 1

Matthijs Coster, Oct 18 2014

nonn

For other primes after a few exceptions it seems that all denominators of harmonic numbers are divisible by that prime. For 11 there are many more exceptions. Maybe infinitely many?

			33 is in the sequence since H(33) = p/q and 11 is not a divisor of q. Here H(n) = Sum_{i=1..n} 1/i.
Of course if H(33) has no denominator with a factor 11 the same is true for 34, 35, ..., 43.

Cf. A002805.

lista(nn) = {forstep (n=0, nn, 11, if (denominator(sum(k=2,n,1/k)) % 11, print1(n, ", ")););} \\ Michel Marcus, Oct 19 2014

n = 10000
sum11 = 0
resu = [0]
for i in range(11, n, 11):
    D = (1 / i).partial_fraction_decomposition()[1]
    sum11 += sum(v for v in D if 11.divides(v.denominator()))
    if sum11 >= 1:
        sum11 -= 1
    if sum11 == 0:
        resu.append(i)
resu

cp's OEIS Frontend

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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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).

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A363539 Decimal expansion of Sum_{k>=1} (H(k)^2 - (log(k) + gamma)^2)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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A363540 Decimal expansion of Sum_{k>=1} (H(k)^3 - (log(k) + gamma)^3)/k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and gamma is Euler's constant (A001620).

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A120308 Numerator((p-1)*H(p-1))/p^2 for p = prime(n) > 3, where H(k) is k-th harmonic number A001008(k)/A002805(k).

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A348373 Decimal expansion of Sum_{k>=1} H(k)^2/2^k, where H(k) = A001008(k)/A002805(k) is the k-th harmonic number.

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A360029 Consider a ruler composed of n segments with lengths 1, 1/2, 1/3, ..., 1/n with total length A001008(n)/A002805(n). a(n) is the minimum number of distinct distances of all pairs of marks that can be achieved by permuting the positions of the segments.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Extensions

A370742 Decimal expansion of Sum_{k>=2} H(k-1) * F(k) / (k*2^k), where H(k) = A001008(k)/A002805(k) is the k-th harmonic number and F(k) = A000045(k) is the k-th Fibonacci number.

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