A034602 -id:A034602 - cp's OEIS frontend

A260210 A034602(n) modulo prime(n).

1, 5, 1, 1, 3, 9, 13, 11, 1, 11, 34, 33, 31, 38, 58, 56, 24, 35, 62, 38, 23, 27, 96, 84, 2, 66, 106, 74, 10, 31, 8, 34, 58, 26, 26, 144, 150, 140, 167, 137, 31, 107, 78, 157, 1, 103, 165, 97, 111, 60, 196, 48, 97, 259, 155, 175, 244, 13, 269, 34, 184, 222, 54

Offset: 3

Felix Fröhlich, Jul 19 2015

nonn

p is a Wolstenholme prime (A088164) iff a(n) = 0. This holds for n = 1944 and n = 157504.
When performing a search for Wolstenholme primes, one can choose an integer constant c >= 0 and record all primes with a(n) <= c in order to get a larger data set.
The values here appear to have a nicer asymptotic growth behavior than those in A260209.
It appears that A260209(n)/a(n) = A001248(n).
The formula only returns integers for primes greater than 3. - Robert G. Wilson v, Jul 29 2015

Felix Fröhlich, Table of n, a(n) for n = 3..10000

Cf. A034602, A088164, A241014, A244801, A250406, A258367, A260209.

f[n_] := Block[{p = Prime@ n}, (Mod[ Binomial[2p - 1, p - 1], p^4] - 1)/p^3]; Array[f, 60, 3] (* Robert G. Wilson v, Jul 29 2015 *)

a(n) = p=prime(n); lift(Mod(binomial(2*p-1, p-1)\p^3, p))

A034602(n)/prime(n) = A260209(n)/prime(n)^2, for n>2. - Robert G. Wilson v, Jul 29 2015

A079681 Duplicate of A034602.

Original entry on oeis.org

1, 5, 265, 2367, 237493, 2576561, 338350897, 616410400171, 7811559753873

Offset: 5

dead

A086656 Duplicate of A034602.

Original entry on oeis.org

1, 5, 265, 2367, 237493, 2576561, 338350897, 616410400171, 7811559753873

Offset: 5

dead

A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

Original entry on oeis.org

16843, 2124679

Offset: 1

Christian Schroeder, Sep 21 2003

McIntosh and Roettger showed that the next term, if it exists, must be larger than 10^9. - Felix Fröhlich, Aug 23 2014
When cb(m)=binomial(2m,m) denotes m-th central binomial coefficient then, obviously, cb(a(n))=2 mod a(n)^4. I have verified that among all naturals 1A246134). One might therefore wonder whether this is true in general. - Stanislav Sykora, Aug 26 2014
Romeo Mestrovic, Congruences for Wolstenholme Primes, Lemma 2.3, shows that the criterion for p to be a Wolstenholme prime is equivalent to p dividing A027641(p-3). In 1847 Cauchy proved that this was a necessary condition for the failure of the first case of Fermat's Last Theorem for the exponent p (see Ribenboim, 13 Lectures, p. 29). - John Blythe Dobson, May 01 2015
Primes p such that p^3 divides A001008(p-1) (Zhao, 2007, p. 18). Also: Primes p such that (p, p-3) is an irregular pair (cf. Buhler, Crandall, Ernvall, Metsänkylä, 1993, p. 152). Keith Conrad observes that for the two known (as of 2015) terms ord_p(H_p-1) = 3 is satisfied, where ord_p(H_p-1) gives the p-adic valuation of H_p-1 (cf. Conrad, p. 5). Romeo Mestrovic conjectures that p is a Wolstenholme prime if and only if S_(p-2)(p) == 0 (mod p^3), where S_k(i) denotes the sum of the k-th powers of the positive integers up to and including (i-1) (cf. Mestrovic, 2012, conjecture 2.10). - Felix Fröhlich, May 20 2015
Primes p that divide the Wolstenholme quotient W_p (A034602). Also, primes p such that p^2 divides the Babbage quotient b_p (A263882). - Jonathan Sondow, Nov 24 2015
The only known composite numbers n such that binomial(2n-1, n-1) is congruent to 1 mod n^2 are the numbers n = p^2 where p is a Wolstenholme prime: see A267824. - Jonathan Sondow, Jan 27 2016
The converse of Wolstenholme's theorem implies that if an integer n satisfies the congruence binomial(2*n-1, n-1) == 1 (mod n^4), then n is a term of this sequence, i.e., then n is necessarily prime, or, equivalently, A298946(i) > 1 for all i > 0. Whether this is true for all such n is an open problem. - Felix Fröhlich, Feb 21 2018
Primes p such that binomial(2*p-1, p-1) == 1-2*p*Sum_{k=1..p-1} 1/k - 2*p^2*Sum_{k=1..p-1} 1/k^2 (mod p^7) (cf. Mestrovic, 2011, Corollary 4). - Felix Fröhlich, Feb 21 2018
These are primes p such that p^2 divides A007406(p-1) (Mestrovic, 2015, p. 241, Lemma 2.3). - Amiram Eldar and Thomas Ordowski, Jul 29 2019
If a third Wolstenholme prime exists it is larger than 6*10^10 (cf. Hathi, Mossinghoff, Trudgian, 2021). - Felix Fröhlich, Apr 27 2021
Named after the English mathematician Joseph Wolstenholme (1829-1891). - Amiram Eldar, Jun 10 2021
Primes p such that tanh(Sum_{k=1..p-1} artanh(k/p)) == 0 (mod p^4). - Thomas Ordowski, Apr 17 2025

Richard K. Guy, Unsolved Problems in Number Theory, Sect. B31.
Paulo Ribenboim, 13 Lectures on Fermat's Last Theorem (Springer, 1979).
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 23.

Ronald Bruck, Wolstenholme's Theorem, Stirling Numbers, and Binomial Coefficients.
Joe Buhler, Richard Crandall, Reijo Ernvall and Tauno Metsänkylä, Irregular primes and cyclotomic invariants to four million, Math. Comp., Vol. 61, No. 203 (1993), pp. 151-153.
Chris Caldwell, The Prime Glossary, Wolstenholme prime.
Leonardo Carofiglio, Luigi De Filpo, and Alessandro Gambini, p-adic valuation of harmonic sums and their connections with Wolstenholme primes, arXiv:2303.15010 [math.NT], 2023.
Keith Conrad, The p-adic growth of harmonic sums.
Shehzad Hathi, Michael J. Mossinghoff, and Timothy S. Trudgian, Wolstenholme and Vandiver primes, The Ramanujan Journal, (2021); arXiv version, 2101.11157 [math.NT], 2021.
Richard J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica, Vol. 71, No. 4 (1995), pp. 381-389.
Richard J. McIntosh and Eric L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. Vol 76, No. 260 (2007), pp. 2087-2094.
R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv:1111.3057 [math.NT], 2011.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
Romeo Meštrović, Congruences for Wolstenholme Primes, arXiv:1108.4178 [math.NT], 2011.
Romeo Meštrović, Congruences for Wolstenholme Primes, Czechoslovak Mathematical Journal, Vol. 65 (2015), pp. 237-253.
Romeo Meštrović, A congruence modulo n^3 involving two consecutive sums of powers and its applications, arXiv:1211.4570 [math.NT], 2012.
Romeo Meštrović, Several generalizations and variations of Chu-Vandermonde identity, arXiv:1807.10604 [math.CO], 2018.
Jonathan Sondow, Extending Babbage's (non-)primality tests, in Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, CANT 2015 and 2016, New York, 2017, pp. 269-277; arXiv:1812.07650 [math.NT], 2018.
Eric Weisstein's World of Mathematics, Wolstenholme Prime.
Eric Weisstein's World of Mathematics, Integer Sequence Primes.
Wikipedia, Wolstenholme prime.
Jianqiang Zhao, Bernoulli numbers, Wolstenholme's theorem, and p^5 variations of Lucas' theorem, J. Number Theory, Vol. 123 (2007), pp. 18-26.

Cf. A000984, A001008, A007406, A027641, A034602, A099908, A246130, A246132, A246133, A246134, A263882, A267824, A298946.

[p: p in PrimesUpTo(2*10^4)| (Binomial(2*p-1,p-1) mod (p^4)eq 1)]; // Vincenzo Librandi, May 02 2015

For[i = 2, i <= 20000, i++, {If[PrimeQ[i] && Mod[Binomial[2*i - 1, i - 1], i^4] == 1, Print[i]]}] (* Dylan Delgado, Mar 02 2021 *)

forprime(n=2, 10^9, if(Mod(binomial(2*n-1, n-1), n^4)==1, print1(n, ", "))); \\ Felix Fröhlich, May 18 2014

A000984(a(n)) = 2 mod a(n)^4. - Stanislav Sykora, Aug 26 2014
A099908(a(n)) == 1 mod a(n)^4. - Jonathan Sondow, Nov 24 2015
A034602(PrimePi(a(n))) == 0 mod a(n) and A263882(PrimePi(a(n))) == 0 mod a(n)^2. - Jonathan Sondow, Dec 03 2015

A010763 a(n) = binomial(2n+1, n+1) - 1.

Original entry on oeis.org

0, 2, 9, 34, 125, 461, 1715, 6434, 24309, 92377, 352715, 1352077, 5200299, 20058299, 77558759, 300540194, 1166803109, 4537567649, 17672631899, 68923264409, 269128937219, 1052049481859, 4116715363799, 16123801841549, 63205303218875, 247959266474051

Offset: 0

N. J. A. Sloane

(With a different offset:) p divides a(p) for prime p. p^2 divides a(p) for prime p > 2. p^3 divides a(p) for prime p > 3 (implied by Wolstenholme's theorem). Wolstenholme's quotients are listed in A034602(n) = a(prime(n))/prime(n)^3 = {1, 5, 265, 2367, 237493, 2576561, 338350897, ...} = a(p)/p^3 for prime p > 3. p^3 divides a(p^k) for prime p > 3 and integer k > 0. Primes in a(n) are listed in A112862(n) = {2, 461, 92377, 269128937219, ...} Primes of the form (2*n)!/(2*(n!)^2) - 1. Numbers n such that a(n) is prime are listed in A112861(n) = {2, 6, 10, 21, 45, 63, 306, 404, 437, 471, 646, ...}. - Alexander Adamchuk, Jan 05 2007

a(n-1) is the number of weak compositions of n into n parts in which at least one part is zero. a(3)=34 since 4 can be written as 4+0+0+0 (4 such compositions); 3+1+0+0 (12 such compositions); 2+2+0+0 (6 such compositions); 2+1+1+0 (12 such compositions). All these weak compositions contain at least one zero. - Enrique Navarrete, Jan 09 2022

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Eric Weisstein's World of Mathematics, Wolstenholme's Theorem
Jianqiang Zhao, Uniform Approach to Double Shuffle and Duality Relations of Various q-Analogs of Multiple Zeta Values via Rota-Baxter Algebras, arXiv preprint arXiv:1412.8044 [math.NT], 2014.

Cf. A001700, A001701.

Cf. A001008, A007406, A112861, A112862, A034602.

[Binomial(2*n-1,n-1)-1: n in [1..30]]; // Vincenzo Librandi, Mar 21 2013

A010763:=n->binomial(2*n+1, n+1) - 1: seq(A010763(n), n=0..30); # Wesley Ivan Hurt, Sep 05 2015

Table[Binomial[2n - 1, n - 1] - 1, {n, 20}] (* Alonso del Arte, Dec 13 2012 *)
CoefficientList[Series[Exp[2*x]*(BesselI[0,2*x] + BesselI[1,2*x]) - Exp[x], {x, 0, 20}], x]*Table[n!, {n, 0, 20}] (* Stefano Spezia, Dec 02 2018 *)

a(n) = binomial(2*n+1, n+1) - 1;
vector(30, n, a(n-1)) \\ Michel Marcus, Sep 05 2015

first(n) = x='x+O('x^n); Vec((1 - sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)) - 1/(1 - x), -n) \\ Iain Fox, Dec 19 2017 (corrected by Iain Fox, Oct 24 2018)

a(n) = (n/(2n+2))*Sum_{k = 1..n+1} C(2n+2, k)/C(n+1, k). - Benoit Cloitre, Aug 20 2002
a(n) = Sum_{i = 1..n} C(n + i, n). - Benoit Cloitre, Oct 15 2002
a(n + 1) = C(2n - 1, n - 1) - 1. - Alonso del Arte, Dec 15 2012
From Ilya Gutkovskiy, Feb 07 2017: (Start)
O.g.f.: (1 - sqrt(1 - 4*x))/(2*x*sqrt(1 - 4*x)) - 1/(1 - x).
E.g.f.: exp(2*x)*(BesselI(0,2*x) + BesselI(1,2*x)) - exp(x). (End)

A127042 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square.

Original entry on oeis.org

2, 3, 5, 7, 17, 19, 29, 31, 37, 41, 97, 127, 131, 211, 223, 227, 229, 233, 239, 241, 439, 443, 449, 457, 461, 463, 727, 733, 739, 743, 751, 757, 761, 769, 773, 863, 877, 881, 883, 887, 967, 971, 977, 983, 991, 997, 1009, 1013, 1187, 1193, 1201, 1901, 1907, 1913, 1931, 1933

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029.

a = {}; Do[If[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]] == Floor[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]]], AppendTo[a, Prime[x]]], {x, 1, 50}]; a

More terms from Franklin T. Adams-Watters, Jan 21 2012

A127046 Primes p such that denominator of Sum_{k=1..p-1} 1/k^3 is a cube.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 29, 31, 37, 41, 83, 89, 97, 137, 139, 293, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Chai Wah Wu, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^3; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/3)], AppendTo[a, i + 1]]]]; a]; d[2000]
Select[Prime[Range[200]], IntegerQ[Surd[Denominator[Sum[1/k^3, {k,#-1}]], 3]]&] (* Harvey P. Dale, Mar 13 2013 *)

A127047 Primes p such that denominator of Sum_{k=1..p-1} 1/k^4 is a fourth power.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 31, 53, 67, 71, 73, 97, 101, 103, 107, 109, 127, 131, 197, 199, 211, 223, 227, 229, 233, 293, 367, 373, 379, 383, 389, 397, 401, 439, 443, 449, 457, 461, 463, 557, 563, 569, 571, 577, 877, 881, 883, 967, 971, 977, 983, 991, 997

Offset: 1

Artur Jasinski, Jan 03 2007

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..665 from Robert Israel)

Cf. A061002, A034602, A127029, A127042, A127046.

S:= 0: R:= NULL: count:= 0:
for k from 1 while count < 100 do
  S:= S + 1/k^4;
  if isprime(k+1) and surd(denom(S),4)::integer then R:= R,k+1; count:= count+1 fi
od:
R; # Robert Israel, Oct 25 2019

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^4; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/4)], AppendTo[a, i + 1]]]]; a]; d[10000]
Select[Flatten[Position[Denominator[Accumulate[1/Range[1000]^4]],?(IntegerQ[ Surd[ #,4]]&)]],PrimeQ] (* _Harvey P. Dale, Feb 08 2015 *)

A127048 Primes p such that denominator of Sum_{k=1..p-1} 1/k^5 is a fifth power.

Original entry on oeis.org

2, 3, 5, 11, 13, 17, 37, 41, 53, 83, 127, 131, 137, 139, 149, 151, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 733, 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, 811, 853, 857, 859, 863, 877, 881, 883, 887, 929, 967, 1091, 1093, 1097, 1103, 1109, 1151

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042, A127046, A127047.

d[n_] := Module[{}, su = 0; a = {}; For[i = 1, i <= n, i++, su = su + 1/ i^5; If[PrimeQ[i + 1], If[IntegerQ[(Denominator[su])^(1/5)], AppendTo[a, i + 1]]]]; a]; d[2000]

A127043 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is not a square.

Original entry on oeis.org

11, 13, 23, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 101, 103, 107, 109, 113, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421

Offset: 1

Artur Jasinski, Jan 03 2007

nonn

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A061002, A034602, A127029, A127042.

S:= 0: R:= NULL: count:= 0:
for k from 1 while count < 100 do
  S:= S + 1/k^2;
  if isprime(k+1) and not issqr(denom(S)) then
       R:= R,k+1; count:= count+1;
  fi
od:
R; # Robert Israel, Oct 25 2019

a = {}; Do[If[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]] == Floor[Sqrt[Denominator[Sum[1/x^2, {x, 1, Prime[x] - 1}]]]], 1,AppendTo[a, Prime[x]]], {x, 1, 50}]; a

More terms from Robert Israel, Oct 25 2019

cp's OEIS Frontend

A260210 A034602(n) modulo prime(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A079681 Duplicate of A034602.

Views

Author

Keywords

A086656 Duplicate of A034602.

Views

Author

Keywords

A088164 Wolstenholme primes: primes p such that binomial(2p-1,p-1) == 1 (mod p^4).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Formula

A010763 a(n) = binomial(2n+1, n+1) - 1.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A127042 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2 is a square.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Extensions

A127046 Primes p such that denominator of Sum_{k=1..p-1} 1/k^3 is a cube.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

A127047 Primes p such that denominator of Sum_{k=1..p-1} 1/k^4 is a fourth power.

Views

Author

Keywords

Links

Crossrefs

Programs

Maple

Mathematica

A127048 Primes p such that denominator of Sum_{k=1..p-1} 1/k^5 is a fifth power.

Views

Author

Keywords

Links

Crossrefs

Programs