A092103 -id:A092103 - cp's OEIS frontend

A034602 Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).

1, 5, 265, 2367, 237493, 2576561, 338350897, 616410400171, 7811559753873, 17236200860123055, 3081677433937346539, 41741941495866750557, 7829195555633964779233, 21066131970056662377432067, 59296957594629000880904587621, 844326030443651782154010715715

Offset: 3

N. J. A. Sloane

nonn

Equivalently, (binomial(2p,p)-2)/(2*p^3) where p runs through the primes >=5.

The values of this sequence's terms are replicated by conjectured general formula, given in A223886 (and also added to the formula section here) for k=2, j=1 and n>=3. - Alexander R. Povolotsky, Apr 18 2013

			Binomial(10,5)-2 = 250; 5^3=125 hence a(5)=1.

R. K. Guy, Unsolved Problems in Number Theory, Sect. B31.

Robert Israel, Table of n, a(n) for n = 3..263
R. R. Aidagulov and M. A. Alekseyev. On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences 233:5 (2018), 626-634. doi:10.1007/s10958-018-3948-0 arXiv:1602.02632
R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica 71:4 (1995), 381-389.
Romeo Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Jonathan Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, 269-277, CANT 2015 and 2016, New York, 2017; arXiv:1812.07650 [math.NT], 2018.

Cf. A177783 (alternative definition of Wolstenholme quotient), A072984, A092101, A092103, A092193, A128673, A217772, A223886, A263882.

Cf. A268512, A268589, A268590.

[(Binomial(2*p-1,p)-1) div p^3: p in PrimesInInterval(4,100)]; // Vincenzo Librandi, Nov 23 2015

f:= proc(n) local p;
p:= ithprime(n);
(binomial(2*p-1,p)-1)/p^3
end proc:
map(f, [$3..30]); # Robert Israel, Dec 19 2018

Table[(Binomial[2 Prime[n] - 1, Prime[n] - 1] - 1)/Prime[n]^3, {n, 3, 20}] (* Vincenzo Librandi, Nov 23 2015 *)

a(n) = (A088218(p)-1)/p^3 = (A001700(p-1)-1)/p^3 = (A000984(p)-2)/(2*p^3), where p=A000040(n).
a(n) = A087754(n)/2.
a(n) = (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j)) / (k*prime(n)^3) for k=2, j=1, and n>=3. - Alexander R. Povolotsky, Apr 18 2013
a(n) = A263882(n)/prime(n) for n > 2. - Jonathan Sondow, Nov 23 2015
a(n) = numerator(tanh(Sum_{k=1..p-1} artanh(k/p)))/p^3, where p = prime(n) for n >= 3. - Thomas Ordowski, Apr 17 2025

Edited by Max Alekseyev, May 14 2010

More terms from Vincenzo Librandi, Nov 23 2015

A092101 Harmonic primes.

Original entry on oeis.org

5, 13, 17, 23, 41, 67, 73, 79, 107, 113, 139, 149, 157, 179, 191, 193, 223, 239, 241, 251, 263, 277, 281, 293, 307, 311, 317, 331, 337, 349, 431, 443, 449, 461, 467, 479, 487, 491, 499, 503, 541, 547, 557, 563, 569, 593, 619, 653, 683, 691, 709, 757, 769, 787

Offset: 1

T. D. Noe, Feb 20 2004

nonn

For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k), the numerator of the harmonic number H(k). For harmonic primes, J_p contains only the three numbers p-1, (p-1)p and (p-1)(p+1). It has been conjectured that there are an infinite number of these primes and that their density in the primes is 1/e.

Prime p=A000040(n) is in this sequence iff neither H(k) == 0 (mod p), nor H(k) == -A177783(n) (mod p) have solutions for 1 <= k <= p-2. - Max Alekseyev, May 13 2010

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302.
A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

Cf. A092102 (non-harmonic primes), A092103 (size of J_p).

is(p)=my(K=-Mod((binomial(2*p-1, p)-1)/2/p^3,p),H=Mod(0,p));for(k=1,p-2,H+=1/k;if(H==0||H==K,return(0)));1 \\ Charles R Greathouse IV, Mar 16 2014

More terms from Max Alekseyev, May 13 2010

A072984 Least k such that prime(n) appears in the factorization of A001008(k) (the numerator of the k-th harmonic number).

Original entry on oeis.org

2, 4, 6, 3, 12, 16, 18, 22, 13, 30, 17, 40, 13, 46, 22, 58, 10, 66, 70, 72, 78, 82, 88, 11, 100, 102, 106, 25, 112, 126, 130, 5, 138, 148, 150, 156, 162, 166, 71, 178, 180, 190, 192, 196, 38, 210, 222, 22, 228, 232, 238, 240, 250, 66, 262, 33, 58, 276, 280, 282

Offset: 2

Benoit Cloitre, Aug 21 2002

a(n)<=n for n =2,5,14,18,25,29,33,46,49,...

For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k). This sequence gives the smallest elements of J_p. The largest elements of J_p are given by A177734. The sizes of J_p are given by A092103.

T. D. Noe, Table of n, a(n) for n = 2..1000
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_, Feb 10 2016]
A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

Cf. A092101 (harmonic primes), A092102 (non-harmonic primes), A092103 (size of Jp).

A072984[n_] := Module[{p, k, sum},
   p = Prime[n]; k = 1; sum = 1/k;
   While[! Divisible[Numerator[sum], p],
    k++; sum += 1/k];
   Return[k]];
Table[A072984[n], {n, 2, 61}] (* Robert Price, May 01 2019 *)

a(n)=if(n<0,0,s=1; while(numerator(sum(k=1,s,1/k))%prime(n)>0,s++); s)

A092102 Non-harmonic primes: the odd primes not in A092101.

Original entry on oeis.org

3, 7, 11, 19, 29, 31, 37, 43, 47, 53, 59, 61, 71, 83, 89, 97, 101, 103, 109, 127, 131, 137, 151, 163, 167, 173, 181, 197, 199, 211, 227, 229, 233, 257, 269, 271, 283, 313, 347, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 433, 439, 457, 463, 509, 521, 523

Offset: 1

T. D. Noe, Feb 20 2004

nonn

For p = prime(n), Boyd defines Jp to be the set of numbers k such that p divides A001008(k), the numerator of the harmonic number H(k). For harmonic primes, Jp contains only the three numbers p-1, (p-1)p and (p-1)(p+1).

Boyd's paper omits 509.

A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302.

Cf. A092101 (harmonic primes), A092103 (size of Jp).

A177734 Largest k such that prime(n) divides the numerator of the k-th harmonic number (=A001008(k)).

Original entry on oeis.org

22, 24, 102728, 1011849771855214912968404217247, 168, 288, 848874360, 528, 695552, 886725671, 50641, 1680, 2359785, 10776888210, 414839198, 42176361744, 226972, 4488, 9094138358932, 5328, 6240

Offset: 2

Max Alekseyev, May 12 2010

For p = prime(n), Boyd defines J_p to be the set of numbers k such that p divides A001008(k). This sequence gives the largest element of J_p. The smallest element of J_p is given by A072984. The size of J_p is given by A092103.

Term a(23) is too large to include, see b-file. - Max Alekseyev, Apr 04 2025

Max Alekseyev, Table of n, a(n) for n = 2..220
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_, Feb 10 2016]
Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.

Cf. A072984, A092103, A092193.

For p = prime(n) in A092101, a(n) = p^2 - 1.

a(5) computed by Boyd.

a(8)-a(22) from Max Alekseyev, Oct 23 2012

A177783 Wolstenholme quotient of prime p=A000040(n), i.e., such integer m

Original entry on oeis.org

3, 6, 6, 7, 10, 14, 18, 20, 16, 24, 17, 38, 39, 19, 29, 28, 12, 53, 31, 19, 53, 58, 48, 42, 1, 33, 53, 37, 5, 81, 4, 17, 29, 13, 13, 72, 75, 70, 173, 159, 111, 150, 39, 178, 106, 163, 196, 163, 172, 30, 98, 24, 177, 261, 212, 223, 122, 147, 276, 17, 92, 111, 27, 209, 241

Offset: 3

Max Alekseyev, May 13 2010

nonn

a(n) = 0 iff A000040(n) is a Wolstenholme prime (given by A088164).

For n>2 and p=A000040(n), H(p^2-p) == H(p^2-1) == a(n)*p (mod p^2).

David W. Boyd, A p-adic study of the partial sums of the harmonic series, Experimental Math., Vol. 3 (1994), No. 4, 287-302.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, Journal of Number Theory, Volume 123, Issue 1, March 2007, Pages 18-26.

Cf. A034602, A072984, A087754, A092101, A092103, A092193, A128673.

{ a(n) = my(p); p=prime(n); ((binomial(2*p-1,p)-1)/2/p^3)%p }

a(n) = H(p-1)/p^2 mod p = A001008(p-1)/A002805(p-1)/p^2 mod p = A034602(n)/2 mod p = (binomial(2*p-1,p)-1)/(2*p^3) mod p, where p = A000040(n).
a(n) = (-1/3)*B(p-3) mod p, with p=prime(n) and B(n) is the n-th Bernoulli number. - Michel Marcus, Feb 05 2016
a(n) = A087754(n)/4 mod A000040(n).

Edited by Max Alekseyev, May 16 2010

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}}]]

cp's OEIS Frontend

A034602 Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n).

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A092101 Harmonic primes.

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A072984 Least k such that prime(n) appears in the factorization of A001008(k) (the numerator of the k-th harmonic number).

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A092102 Non-harmonic primes: the odd primes not in A092101.

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A177734 Largest k such that prime(n) divides the numerator of the k-th harmonic number (=A001008(k)).

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A177783 Wolstenholme quotient of prime p=A000040(n), i.e., such integer m

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PARI

Formula

Extensions

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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Examples

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Mathematica