A092101 -id:A092101 - cp's OEIS frontend

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

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).

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

Original entry on oeis.org

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

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)

A092103 Number of values of k for which prime(n) divides A001008(k), the numerator of the k-th harmonic number.

Original entry on oeis.org

3, 3, 13, 638, 3, 3, 25, 3, 18, 26, 15, 3, 27, 24, 17, 23, 13, 3, 45, 3, 3, 43038, 7, 74, 44, 63, 3, 1273, 3, 3515, 7, 38, 3, 3, 7, 3, 74, 526, 288, 3, 19, 3, 3, 41, 11, 59, 3, 31, 65, 176, 3, 3, 3, 20, 3, 106, 55, 3, 3, 89, 3, 3, 3, 79, 3, 3, 3, 47, 3, 21, 253, 29, 7, 79, 41, 19, 701533, 13, 9, 703, 23, 3, 205, 105, 3, 3, 323, 3, 7, 3, 3, 3, 3, 3, 3, 13, 1763

Offset: 2

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). This sequence gives the size of J_p. A072984 and A177734 give the smallest and largest elements of J_p, respectively.
A092101 gives primes prime(n) such that a(n) = 3 (i.e., a(A000720(A092101(m))) = 3 for all m). A092102 gives primes prime(n) such that a(n) > 3.
From Carlo Sanna, Apr 06 2016: (Start)
Eswarathasan and Levine conjectured that for any prime number p the set J_p is finite.
I proved that if J_p(x) is the number of integers in J_p that are less than x > 1, then J_p(x) < 129 p^(2/3) x^0.765 for any prime p. In particular, J_p has asymptotic density zero. (End)
Bing-Ling Wu and Yong-Gao Chen improved Sanna's (see previous comment) result showing that J_p(x) <= 3 x^(2/3 + 1/(25 log p)) for any prime p and any x > 1. - Carlo Sanna, Jan 12 2017

			a(2) = 3 because 3 divides A001008(k) for k = 2, 7, and 22.
a(4) = 13 because 7 divides A001008(k) for only the 13 values k = 6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735, and 102728. This is the 4th row in A229493.

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_, Oct 23 2012]
Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.
A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.
Carlo Sanna, On the p-adic valuation of harmonic numbers, J. Number Theory 166 (2016), 41-46.
Bing-Ling Wu and Yong-Gao Chen, On certain properties of harmonic numbers, J. Number Theory 175 (2017), 66-86.

Cf. A072984, A092101, A092102.
Cf. A092193 (number of generations for each prime).
Cf. A229493 (terms for each prime).

a(8), a(15), and a(17) corrected by Max Alekseyev, Oct 23 2012

Terms a(23) onward from Carofiglio et al. (2025) added by Max Alekseyev, Apr 01 2025

A092193 Number of generations for which prime(n) divides A001008(k) for some k.

Original entry on oeis.org

4, 3, 7, 30, 3, 3, 8, 3, 5, 7, 4, 3, 5, 7, 6, 7, 4, 3, 8, 3, 3, 339, 4, 11, 10, 14, 3, 47, 3, 146, 4, 8, 3, 3, 4, 3, 20, 49, 33, 3, 6, 3, 3, 11, 5, 12, 3, 6, 17, 21, 3, 3, 3, 5, 3, 20, 18, 3, 3, 14, 3, 3, 3, 11, 3, 3, 3, 10, 3, 6, 35, 8, 4, 13, 11, 8, 1815, 5, 4, 52, 5, 3, 30, 11, 3, 3, 36, 3, 4, 3, 3, 3, 3, 3, 3, 4, 61, 4, 3, 3, 3, 3, 3, 8, 28, 4, 3, 6, 4, 6, 21, 19, 3, 94

Offset: 2

T. D. Noe, Feb 24 2004; corrected Jul 28 2004

nonn

For any prime p, generation m consists of the numbers p^(m-1) <= k < p^m. The zeroth generation consists of just the number 0. When there is a k in generation m such that p divides A001008(k), then that k may generate solutions in generation m+1. It is conjectured that for all primes there are solutions for only a finite number of generations.

Boyd's table 3 states incorrectly that harmonic primes have 2 generations; harmonic primes have 3 generations.

			a(4)=7 because the fourth prime, 7, divides A001008(k) for k = 6, 42, 48, 295, 299, 337, 341, 2096, 2390, 14675, 16731, 16735 and 102728. These values of k fall into 6 generations; adding the zeroth generation makes a total of 7 generations.

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_, Apr 01 2025]
Leonardo Carofiglio, Giacomo Cherubini, and Alessandro Gambini, On Eswarathasan--Levine and Boyd's conjectures for harmonic numbers, arXiv:2503.15714 [math.NT], 2025.
A. Eswarathasan and E. Levine, p-integral harmonic sums, Discrete Math. 91 (1991), 249-257.

Cf. A072984 (least k such that prime(n) divides A001008(k)), A092101 (harmonic primes), A092102 (non-harmonic primes).

a(8), a(15), a(17) corrected, and terms a(23) onward from Carofiglio et al. (2025) added by Max Alekseyev, Apr 01 2025

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

A092195 Primes p that do not divide A001008(k), the numerator of the k-th harmonic number H(k), for any k < p-1.

Original entry on oeis.org

3, 5, 7, 13, 17, 19, 23, 31, 41, 47, 59, 67, 71, 73, 79, 83, 89, 101, 103, 107, 113, 127, 131, 139, 149, 151, 157, 163, 167, 179, 181, 191, 193, 197, 211, 223, 229, 233, 239, 241, 251, 263, 277, 281, 283, 293, 307, 311, 317, 331, 337, 349, 359, 367, 373, 383

Offset: 1

T. D. Noe, Feb 24 2004

nonn

Harmonic primes A092101 are a subset of these primes. Because these primes are analogous to the regular primes A007703 that divide the numerators of Bernoulli numbers, they might be called H-regular primes. The density of these primes is about 0.6 -- very close to the density of regular primes.

Eric Weisstein's World of Mathematics, Harmonic Number
Eric Weisstein's World of Mathematics, Regular Prime

Cf. A072984 (least k such that prime(n) divides A001008(k)).

n=1; Table[While[cnt=0; n++; p=Prime[n]; k=1; h=0; While[k<=(p-1)/2, h=h+1/k; If[Mod[Numerator[h], p]==0, cnt++ ]; k++ ]; cnt>0, ]; p, {100}]

A097279 Alternating-harmonic primes.

Original entry on oeis.org

5, 11, 23, 59, 67, 83, 89, 101, 107, 109, 127, 163, 167, 197, 229, 233, 251, 283, 311, 317, 349, 421, 491, 557, 577, 643, 673, 683, 719, 727, 761, 827, 1009, 1061, 1129, 1163, 1193, 1231, 1327, 1373

Offset: 1

T. D. Noe, Aug 04 2004

These primes, analogous to the harmonic primes in A092101, divide exactly one term of A058313, the numerators of the alternating harmonic numbers. It can be shown that for prime p > 3, if p = 6k-1, then p divides A058313(4k-1), otherwise if p = 6k+1, then p divides A058313(4k). Much of the analysis by Eswarathasan and Levine applies to alternating harmonic sums.

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

maxPrime=1000; lst={}; Do[p=Prime[n]; cnt=0; s=0; i=1; While[s=s+(-1)^(i-1)/i; If[Mod[Numerator[s], p]==0, cnt++ ]; cnt<2&&i

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

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

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A092103 Number of values of k for which prime(n) divides A001008(k), the numerator of the k-th harmonic number.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

A092193 Number of generations for which prime(n) divides A001008(k) for some k.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Formula

Extensions

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

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A092195 Primes p that do not divide A001008(k), the numerator of the k-th harmonic number H(k), for any k < p-1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs