A131657 -id:A131657 - cp's OEIS frontend

A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j and B_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j * (Sum__{k=j+1..jn} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)A_n(z))) has integral coefficients. The sequence is u(n).

1, 1, 1, 2, 2, 36, 36, 144, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 1567641600, 1567641600, 156764160000, 49380710400000, 217275125760000, 1086375628800000, 1738201006080000

Offset: 1

Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007, Sep 30 2007

nonn

Different from A131657 and A056612.

Christian Krattenthaler, Table of n, a(n) for n = 1..40
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151(2) (2010), 175-218.

Cf. A007757 (bisection at even integers), A056612, A131657.

A formula, conditional on a widely believed conjecture, can be found in the article by Krattenthaler and Rivoal (2007-2009) cited in the references: see Theorem 4 and the accompanying remarks.

A007757 Dwork-Kontsevich sequence evaluated at 2*n.

Original entry on oeis.org

1, 2, 36, 144, 1440, 17280, 241920, 29030400, 1567641600, 156764160000, 217275125760000, 1738201006080000, 45193226158080000, 3796230997278720000, 113886929918361600000, 1822190878693785600000, 22489479824838701875200000, 28336744579296764362752000000, 1076796294013277045784576000000, 1679802218660712191423938560000000

Offset: 1

Richard E. Borcherds (reb(AT)math.berkeley.edu)

nonn

For n positive, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum_{k=j+1..j*n} (1/k)), and let b(n) be the largest integer for which exp(B_n(z)/(b(n)*A_n(z))) has integral coefficients. The sequence is b(2*n).

A formula, conditional on a widely believed conjecture, can be found in the Krattenthaler-Rivoal (2007-2009) paper; see Theorem 4 with k = 1 and the remarks on the top of page 8. Since R. E. Borcherds defined a sequence b(n), but then only entered b(2*n) in the OEIS, the formula has to be taken with n replaced by 2*n. - Christian Krattenthaler (Christian.Krattenthaler(AT)univie.ac.at), Sep 12 2007

			G.f. = x + 2*x^2 + 36*x^3 + 144*x^4 + 1440*x^5 + 17280*x^6 + 241920*x^7 + ...

Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, 3(3) (2009), 555-591. [Part II appeared before Part I.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, Duke Math. J. 151(2) (2010), 175-218.

Cf. A131657, A131658, A056612.

a[n0_] := Module[{A, MM = 2, n = 2n0, c1, c2}, A = Exp[Sum[x^j (n j)!/ (j!^n) Sum[1/k, {k, j+1, j n}], {j, 0, MM}]/Sum[x^j (n j)!/(j!^n), {j, 0, MM}]]; c1 = SeriesCoefficient[A, {x, 0, 1}]; c2 = SeriesCoefficient[A, {x, 0, 2}]; GCD[c1, (c1 + c1^2)/2 - c2]];
Array[a, 20] (* Jean-François Alcover, Dec 17 2018, from PARI *)

{a(n) = my(A, MM=2, c1, c2); if(n<1, 0, n*=2; A = x * O(x^MM); A = exp( sum(j=0, MM, x^j * (n*j)! / (j!^n) * sum(k=j+1, j*n, 1/k), A) / sum(j=0, MM, x^j * (n*j)! / (j!^n), A)); c1 = polcoeff(A, 1); c2 = polcoeff(A, 2); gcd(c1, (c1 + c1^2)/2 - c2))}; /* Michael Somos, Nov 16 2006 */

Definition in comment line, PARI code and terms of sequence corrected by Christian Krattenthaler (christian.krattenthaler(AT)univie.ac.at), Sep 30 2007

a(8) corrected by Sean A. Irvine, Jan 22 2018

A056612 a(n) = gcd(n!, n!*(1 + 1/2 + 1/3 + ... + 1/n)).

Original entry on oeis.org

1, 1, 1, 2, 2, 36, 36, 144, 144, 1440, 1440, 17280, 17280, 241920, 3628800, 29030400, 29030400, 1567641600, 1567641600, 156764160000, 9876142080000, 217275125760000, 217275125760000, 1738201006080000, 1738201006080000

Offset: 1

Leroy Quet, Aug 08 2000

The first difference between this sequence and A131657 occurs for n = 20, while the first difference between this sequence and A131658 occurs for n = 21. - Christian Krattenthaler, Sep 30 2007

			a(4) = gcd(4!, 4!*(1 + 1/2 + 1/3 + 1/4)) = gcd(24, 50) = 2.
a(4) = gcd(A000254(5), A000254(4)) = gcd(5!*(1 + 1/2 + 1/3 + 1/4 + 1/5), 4!*(1 + 1/2 + 1/3 + 1/4)) = gcd(274, 50) = 2. - _Petros Hadjicostas_, May 18 2020

Michael De Vlieger, Table of n, a(n) for n = 1..535

Cf. A000142, A001008, A002805.
Cf. A007757, A131657, A131658.
Cf. A334958 (similar sequence for the alternative harmonic series).

Table[GCD[#, # Total@ Map[1/# &, Range@ n]] &[n!], {n, 25}] (* Michael De Vlieger, Sep 23 2017 *)
a[n_] := n!/Denominator@ HarmonicNumber@ n; Array[a, 25] (* Robert G. Wilson v, Jun 30 2018 *)

a(n) = gcd(n!, n!*sum(k=1, n, 1/k)); \\ Michel Marcus, Jul 14 2018

a(n) = gcd(stirling(n+1, 2, 1), n!); \\ Michel Marcus, May 20 2020

a(n) = A000142(n)/A002805(n) = A000254(n)/A001008(n). - Franz Vrabec, Sep 13 2005
a(n) = gcd(A000254(n+1), A000254(n)). - Petros Hadjicostas, May 18 2020
a(n) = gcd(Stirling1(n+1, 2), n!). - Michel Marcus, May 20 2020

A268112 Numbers k for which the numerator of the k-th harmonic number H_k is divisible by the third power of a prime less than k.

Original entry on oeis.org

848, 9338, 10583, 3546471722268916272

Offset: 1

Felix Fröhlich, Jan 26 2016

The sequence contains numbers k for which there is a prime p < k with v_p(H_k) >= 3, where v_p(x) is the p-adic valuation of x and H_k is the k-th Harmonic number. All numbers were found by D. W. Boyd. The corresponding p for a(1) through a(4) is 11 while for a(5) (in the b-file) is 83. [Edited by Petros Hadjicostas, May 25 2020]
It is a widely believed conjecture that there is no pair of an integer k and a prime p for which v_p(H_k) >= 4. If variations of this conjecture hold, then Krattenhaler and Rivoal (2007-2009) would be able to establish some formulas for their theory. See also A007757, A131657, and A131658. - Petros Hadjicostas, May 25 2020
Terms a(4) and a(5) are conjectural as possible existence of smaller terms is not eliminated. Carofiglio et al. (2025) computed some further terms (see links). - Max Alekseyev, Apr 01 2025

Petros Hadjicostas, Table of n, a(n) for n = 1..5
David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.
L. Carofiglio and G. Cherubini, A copy of the supplementary file with some large terms from Carofiglio et al. (2025) study.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 73-94.

Cf. A001008, A007757, A131657, A131658.

h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n-1, if(valuation((numerator(h(n))), p) > 2, return(1))); return(0)} \\ Edited by Petros Hadjicostas, May 25 2020

Name edited by and a(5) copied from the references by Petros Hadjicostas, May 25 2020

A335189 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 0, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

Original entry on oeis.org

21, 23, 43, 47, 66, 68, 78, 82, 86, 111, 115, 119, 157, 160, 164, 167, 273, 287, 343, 359, 438, 442, 456, 460, 507, 527, 579, 581, 615, 665, 813, 818, 834, 839, 931, 943, 947, 959, 1082, 1090, 1111, 1119, 1140, 1148, 1248, 1288, 1333, 1340, 1346, 1354, 1360, 1367, 1592, 1640, 1641, 1679, 1807, 1847, 2034, 2067, 2069, 2163, 2190

Offset: 1

Petros Hadjicostas, May 26 2020

nonn

For more numbers in this list (up to 10^6), see one of the links below by Krattenthaler and Rivoal. The first few numbers L for which v_p(H_L-1) = 2 (rather than 1) for some prime p <= L are 43, 2034 and 2069 with corresponding primes 7, 13 and 7.

The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.

David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H_L - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H_L - 1) > 0.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
Carlo Sanna, On the p-adic valuation of harmonic numbers, Journal of Number Theory, 166 (2016), 41-46.

Cf. A007757, A131657, A131658, A268112, A334981.

h(n) = sum(i=1, n, 1/i);
is(n) = {forprime(p=1, n, if(valuation((numerator(h(n)-1)), p) > 0, return(1))); return(0)};
for(n=1, 1000, if(is(n)==1, print1(n,",")))

A334981 Numbers k such that the numerator of the k-th alternating harmonic number H'(k) is divisible by the square of a prime less than k.

Original entry on oeis.org

30, 241, 1057, 1499, 10499, 318425

Offset: 1

Petros Hadjicostas, May 25 2020

This sequence was inspired by unsolved conjectures related to the papers by Boyd (1994) and Krattenhaler and Rivoal (2007-2009, 2009) about the harmonic numbers H(k) = Sum_{i=1..k} 1/i. See also the comments for sequences A007757, A131657, A131658, and A268112. Here we are dealing with the alternating harmonic numbers H'(k) = Sum_{i=1..k} (-1)^(i+1)/i.
For the harmonic numbers H(k), it is not known whether there is k >= 1 and a prime p such that v_p(H(k)) >= 4, where v_p(x) is the p-adic valuation of x. Since p cannot be present in both the numerator and the denominator of H(k), this is equivalent to saying that the numerator of H(k) cannot be divisible by the fourth power of a prime p.
If variations of the above conjecture are true, then some conditional results in Krattenhaler and Rivoal (2007-2009, 2009) would hold. Boyd (1994) found only 5 integers k such that there is a prime p < k with v_p(H(k)) >= 3. Since 1994 no other k's have been found that satisfy the latter inequality.
We claim that a similar conjecture holds for the alternating harmonic numbers H'(k): there is no pair of an integer k and a prime p such that v_p(H'(k)) >= 4; i.e., there is no k for which the numerator of H'(k) is divisible by the fourth power of a prime.
This sequence contains those k's for which there is a prime p < k with v_p(H'(k)) >= 2. Up to 2000, we have only been able to find four such k's. The corresponding primes for 30, 241, 1057, and 1499 are 7, 19, 37, and 7. We have v_7(H'(30)) = v_19(H'(241)) = v_37(H'(1057)) = 2, while v_7(H'(1499)) = 3.
It holds v_7(H'(10499)) = 2 and  v_691(H'(318425)) = 2. a(7) > 5*10^5. - Giovanni Resta, May 26 2020

David W. Boyd, A p-adic study of the partial sum of the harmonic series, Experimental Mathematics, 3(4) (1994), 287-302.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.
Tamás Lengyel, On p-adic properties of the Stirling numbers of the first kind, Journal of Number Theory, 148 (2015), 73-94.

Cf. A007757, A131657, A131658, A268112 (similar sequence for harmonic numbers).

ah(n) = sum(i=1, n, (-1)^(i+1)/i);
is(n) = {forprime(p=1, n-1, if(valuation((numerator(ah(n))), p) > 1, return(1))); return(0)}

listaa(nn) = {my(h=0,s=1,nh); for (n=1, nn, h += s/n; nh = numerator(h); forprime(p=1, n-1, if(valuation(nh, p) > 1, print1(n, ", "); break)); s = -s;);} \\ Michel Marcus, May 26 2020

a(5) from Michel Marcus, May 26 2020

a(6) from Giovanni Resta, May 26 2020

A335207 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

Original entry on oeis.org

43, 2034, 2069, 9702, 9712, 67258, 102691, 102727, 147253, 904332

Offset: 1

Petros Hadjicostas, May 26 2020

This is a subset of A335189. All numbers in this list were copied from one of the links below by Krattenthaler and Rivoal.
For all L in this list (up to 904332), we have v_p(H_L - 1) = 2 with corresponding primes as follows: p(1) = 7, p(2) = 13, p(3) = 7, p(4) = p(5) = 11, p(6) = 41, p(7) = p(8) = 11, p(9) = 53, and p(10) = 97.
The calculation of v_p(H_L-1) and v_p(H_L) for all primes p <= L is related to some results about the integrality of the Taylor coefficients of mirror maps. See Theorems 3 and 4 in Krattenthaler and Rivoal (2007-2009, 2009) and sequences A007757, A131657, and A131658.

Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, arXiv:0709.1432 [math.NT], 2007-2009.
Christian Krattenthaler and Tanguy Rivoal, Supplement 2 to the paper "On the integrality of the Taylor coefficients of mirror maps", 2007-2009. [This table contains all triplets of numbers (L, p, v_p(H_L - 1)) such that 1 <= L <= 10^6, p prime <= L, and v_p(H_L - 1) > 0.]
Christian Krattenthaler and Tanguy Rivoal, On the integrality of the Taylor coefficients of mirror maps, II, Communications in Number Theory and Physics, Volume 3, Number 3 (2009), 555-591.

Cf. A007757, A131657, A131658, A268112, A335189.

A335207_list := proc(bound) local p, h, H, L, n;
L := NULL; h := 0;
for n from 1 to bound do
    h := h + 1/n; H := h - 1; p:= 2;
    while p <= n do
        if padic:-ordp(H, p) <= 1
           then p := nextprime(p);
           else L := L, n; break;
        fi
    od;
od; L end:
A335207_list(2222); # Peter Luschny, May 29 2020

list(nn) = {my(h=-1); for (n=1, nn, h += 1/n; forprime(p=1, n-1, if(valuation(h, p) > 1, print1(n, ", "); break)););} \\ Petros Hadjicostas, May 26 2020, courtesy of Michel Marcus

cp's OEIS Frontend

A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j and B_n(z) = Sum_{j>=0} (n*j)!/(j!^n) * z^j * (Sum__{k=j+1..j*n} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)*A_n(z))) has integral coefficients. The sequence is u(n).

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A007757 Dwork-Kontsevich sequence evaluated at 2*n.

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A056612 a(n) = gcd(n!, n!*(1 + 1/2 + 1/3 + ... + 1/n)).

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A268112 Numbers k for which the numerator of the k-th harmonic number H_k is divisible by the third power of a prime less than k.

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A335189 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 0, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

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A334981 Numbers k such that the numerator of the k-th alternating harmonic number H'(k) is divisible by the square of a prime less than k.

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A335207 Numbers L such that there is a prime p <= L for which v_p(H_L - 1) > 1, where v_p(x) is the p-adic valuation of x and H_L is the L-th harmonic number.

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A131658 For n >= 1, put A_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j and B_n(z) = Sum_{j>=0} (nj)!/(j!^n) z^j * (Sum__{k=j+1..jn} (1/k)), and let u(n) be the largest integer for which exp(B_n(z)/(u(n)A_n(z))) has integral coefficients. The sequence is u(n).