A001008 -id:A001008 - cp's OEIS frontend

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).

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).

A093818 a(n) = gcd(A001008(n), n!).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 5, 1, 3, 1, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 11, 1, 1, 1, 11, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Vladeta Jovovic, May 20 2004

Conjecture: every odd prime occurs as a term in the sequence.

Observation: Terms other than 1 are rare. Of the terms a(1) .. a(29524), only 187 are larger than one. Among these 187 terms, the following 50 distinct values occur: 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 227, 257, 269, 509, 863, 919, 1049, 1331, 9409, 11881. Of these, all other are primes except 121 = 11*11, 1331 = 11*11*11, 9409 = 97*97 and 11881 = 109*109. - Antti Karttunen, Aug 28 2017

Antti Karttunen, Table of n, a(n) for n = 1..29524

Cf. A001008, A060746.

A001008(n) = numerator(sum(i=1, n, 1/i)); \\ This function from Michael B. Porter, Dec 08 2009
A093818(n) = gcd(A001008(n),n!); \\ Antti Karttunen, Aug 28 2017

More terms from David Wasserman, Apr 20 2007

Name edited (A001008 substituted for "Wolstenholme") by Antti Karttunen, Aug 28 2017

A120263 Ratio of the numerator of n*HarmonicNumber[n] to the numerator of HarmonicNumber[n]: A096617(n)/A001008(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 25, 1, 1, 1, 1

Offset: 1

Alexander Adamchuk, Jun 26 2006

a(n) is not equal to 1 when n belongs to A074791 - numbers n such that n does not divide the denominator of the n-th harmonic number.
a(n) is almost always equal to 1 except for n=6,18,20,21,33,42,54,.. when a(n) seems to be equal to a prime divisor of n.
a(n) could be equal to a squared prime divisor of n as for n=100,294,500,847,..

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A096617, A001008, A074791.

[Numerator(n*HarmonicNumber(n))/Numerator(HarmonicNumber(n)): n in [1..100]]; // G. C. Greubel, Sep 01 2018

Numerator[Table[n*Sum[1/i,{i,1,n}],{n,1,500}]]/Numerator[Table[Sum[1/i,{i,1,n}],{n,1,500}]]

{h(n) = sum(k=1,n,1/k)};
for(n=1,100, print1(numerator(n*h(n))/numerator(h(n)), ", ")) \\ G. C. Greubel, Sep 01 2018

a(n) = A096617(n)/A001008(n) = numerator[n*Sum[1/i,{i,1,n}]] / numerator[Sum[1/i,{i,1,n}]].
a(n) = n / gcd(denominator(H(n)),n), where H(n) = sum(1/k, k=1..n). [Gary Detlefs, Sep 05 2011]
a(n) = A096617(n)*A110566(n)/A025529(n). [Arkadiusz Wesolowski, Mar 29 2012]

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.

A153357 Numbers n such that the harmonic number numerator A001008(n) is a semiprime.

Original entry on oeis.org

4, 6, 11, 14, 15, 17, 19, 20, 23, 25, 31, 33, 34, 35, 37, 39, 49, 53, 55, 59, 61, 68, 90, 93, 94, 101, 116, 117, 121, 124, 145, 155, 158, 163, 169, 170, 186, 193, 194, 199, 205, 211, 214, 245, 258, 259, 264, 267, 283, 311, 315, 328, 340, 347, 359, 365, 371, 385

Offset: 1

Alexander Adamchuk, Dec 24 2008

414, 421, 425, 436, 451, 452, and 480 are in the sequence. 391 and 476 are the remaining candidates below 500. - Daniel M. Jensen, Jun 26 2020

Numerator(H_391) is fully factored and confirmed semiprime with the help of NFS@Home. - Tyler Busby, May 06 2024

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 259.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, page 347.
D. E. Knuth, The Art of Computer Programming. Addison-Wesley, Reading, MA, Vol. 1, p. 615

Tyler Busby, Table of n, a(n) for n = 1..65
FactorDB, Status of Numerator(H_476) in factordb.com
Hisanori Mishima, Wolstenholme number (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 500, n = 501 to 600).
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem, Harmonic Number.

Cf. A001008 (numerators of harmonic number H(n)=Sum_{i=1..n} 1/i).

Cf. A002805, A056903, A067657.

More terms from Sean A. Irvine, Aug 22 2011
Two missing terms added by D. S. McNeil, Aug 23 2011
More terms from Sean A. Irvine, Apr 01 2013
Two more terms from Daniel M. Jensen, Jun 26 2020

A229493 Irregular triangle in which row n has numbers k such that prime(n) divides A001008(k), the numerator of the k-th harmonic number.

Original entry on oeis.org

2, 7, 22, 4, 20, 24, 6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735, 102728, 3, 7, 10, 77, 80, 84, 87, 110, 113, 117, 120, 848, 852, 856, 882, 888, 958, 962, 966, 1291, 1293, 9328, 9331, 9335, 9338, 9376, 9378, 10583, 10587, 10591, 14205, 14207

Offset: 2

T. D. Noe and Arkadiusz Wesolowski, Nov 11 2013

The length of each row is given in A092103.

			The irregular triangle begins:
2, 7, 22
4, 20, 24
6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735, 102728
3, 7, 10, 77, 80, 84, 87, 110, 113, 117, 120, 848, 852, 856, 882, 888,...

David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302. [WARNING: Table 2 contains miscalculations for p=19, 47, 59, ... - _Max Alekseyev_, Oct 23 2012]
A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.
C. Sanna, On the p-adic valuation of harmonic numbers, J. Number Theory 166 (2016), 41-46.

Cf. A092103 (number of k for which prime(n) divides A001008(k)).

(* rows 2, 3, and part of 4 *) h = ParallelTable[Numerator[HarmonicNumber[i]], {i, 10000}]; Flatten[Table[Position[h, _?(Mod[#, p] == 0 &)], {p, {3, 5, 7}}]]

A309397 a(n) = gcd(n^2, A001008(n-1)) for n > 1.

Original entry on oeis.org

1, 3, 1, 25, 1, 49, 1, 1, 1, 121, 1, 169, 1, 1, 1, 289, 1, 361, 1, 1, 1, 529, 1, 5, 1, 1, 1, 841, 1, 961, 1, 1, 1, 1, 1, 1369, 1, 1, 1, 1681, 1, 1849, 1, 1, 1, 2209, 1, 7, 1, 1, 1, 2809, 1, 1, 1, 1, 1, 3481, 1, 3721, 1, 1, 1, 1, 1, 4489, 1, 1, 1, 5041, 1, 5329

Offset: 2

Amiram Eldar and Thomas Ordowski, Jul 28 2019

nonn

By Wolstenholme's theorem, if p > 3 is prime, then a(p) = p^2.
Conjecture: for n > 3, if a(n) = n^2, then n is a prime.
Note: the weak pseudoprimes n such that a(n) = n are not known.
Composite numbers m <> p^2 for which a(m) > 1 are the same as in A309391: 88, 1290, 9339, ...

			a(11) = gcd(11^2, A001008(11-1)) = gcd(121, 7381) = 121.

R. Meštrović, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862--2012), arXiv:1111.3057 [math.NT], 2011.
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem.
Wikipedia, Wolstenholme's theorem.

Cf. A001008, A007406 (see our comment), A309391.

[Gcd(k^2, Numerator(HarmonicNumber(k-1))):k in [2..80]]; // Marius A. Burtea, Jul 28 2019

a[n_] := GCD[n^2, Numerator[HarmonicNumber[n-1]]]; Array[a, 72, 2]

from sympy import gcd, harmonic
def A309387(n):
    return gcd(n**2,harmonic(n-1).p) # Chai Wah Wu, Jul 31 2019

a(n) = A309391(n) for composite n.
a(p) = p^2 for every prime p > 3.
a(p^2) = p iff p > 3 is a prime.

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

cp's OEIS Frontend

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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A093818 a(n) = gcd(A001008(n), n!).

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A120263 Ratio of the numerator of n*HarmonicNumber[n] to the numerator of HarmonicNumber[n]: A096617(n)/A001008(n).

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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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A153357 Numbers n such that the harmonic number numerator A001008(n) is a semiprime.

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A229493 Irregular triangle in which row n has numbers k such that prime(n) divides A001008(k), the numerator of the k-th harmonic number.

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A309397 a(n) = gcd(n^2, A001008(n-1)) for n > 1.

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