A087754 -id:A087754 - 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

A268589 a(n) = (2C(3p,p) - 9C(2p,p) + 12) / p^5, where p = prime(n).

Original entry on oeis.org

12, 2364, 43500, 20791626, 514377588, 373783661124, 9888937247184828, 312285010312512084, 11167980739981519994382, 13185583459205473525798038, 462369843775374621687338484, 588608385261717115044847555476, 28758863221144089886068560242560564, 1508365481231852329668720928730586740868

Offset: 4

Max Alekseyev, Feb 07 2016

a(n) is an integer for all n>=4, see A268512.

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 preprint, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A268512, A087754, A268590.

Table[(2*Binomial[3p,p]-9*Binomial[2p,p]+12)/p^5,{p,Prime[Range[4,20]]}] (* Harvey P. Dale, Aug 27 2025 *)

{ A268589(n) = my(p=prime(n)); (12 - 9*binomial(2*p,p) + 2*binomial(3*p,p))/p^5; }

A268590 a(n) = (3C(4p,p) - 20C(3p,p) + 54*C(2p,p) - 60) / p^7, where p = prime(n).

Original entry on oeis.org

984, 27780, 32144568, 1269360060, 2470299005220, 316528131552725460, 17262503097511844124, 3329177348896984023277536, 12461979236231507288981559840, 783882118494853605112684502280, 3251723952081272231067929776337100, 959689034437453143807696476144553320100

Offset: 5

Max Alekseyev, Feb 07 2016

nonn

a(n) is an integer for all n>=5, see A268512.

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 preprint, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A268512, A087754, A268589.

{ A268590(n) = my(p=prime(n)); (-60 + 54*binomial(2*p,p) - 20*binomial(3*p,p) + 3*binomial(4*p,p))/p^7; }

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).

Original entry on oeis.org

1, 2, 1, 12, 9, 2, 60, 54, 20, 3, 840, 840, 400, 105, 12, 2520, 2700, 1500, 525, 108, 10, 27720, 31185, 19250, 8085, 2268, 385, 30, 360360, 420420, 280280, 133770, 45864, 10780, 1560, 105, 720720, 864864, 611520, 321048, 127008, 36960, 7488, 945, 56, 12252240, 15036840, 11138400, 6297480, 2776032, 942480

Offset: 1

Max Alekseyev, Feb 06 2016

			n=1: 1
n=2: 2, 1
n=3: 12, 9, 2
n=4: 60, 54, 20, 3
n=5: 840, 840, 400, 105, 12
...
For all primes p>3, p^3 divides 2 - binomial(2*p,p) (cf. A087754).
For all primes p>5, p^5 divides 12 - 9*binomial(2*p,p) + 2*binomial(3*p,p) (cf. A268589).
For all primes p>7, p^7 divides 60 - 54*binomial(2*p,p) + 20*binomial(3*p,p) - 3*binomial(4*p,p) (cf. A268590).

R. R. Aidagulov, 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; also arXiv, arXiv:1602.02632 [math.NT], 2016-2018.

Cf. A099996 (first column), A068550 (diagonal), A087754, A268589, A268590, A254593.

a3418[n_] := LCM @@ Range[n];
c[1, 1] = 1; c[n_, i_] := a3418[2(n-1)] Binomial[2n-1, n-i] ((2i-1)/i/ Binomial[2n-1, n]);
Table[c[n, i], {n, 1, 10}, {i, 1, n}] // Flatten (* Jean-François Alcover, Dec 04 2018 *)

{ A268512(n,i) = lcm(vector(2*(n-1),i,i)) * binomial(2*n-1,n-i) * (2*i-1) / i / binomial(2*n-1,n) }

c(n,i) = A003418(2*(n-1))*binomial(2*n-1,n-i)*(2*i-1)/i/binomial(2*n-1,n).

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

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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A268589 a(n) = (2*C(3p,p) - 9*C(2p,p) + 12) / p^5, where p = prime(n).

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A268590 a(n) = (3*C(4p,p) - 20*C(3p,p) + 54*C(2p,p) - 60) / p^7, where p = prime(n).

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A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^i*c(n,i)*binomial(i*p,p) is divisible by p^(2n-1).

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

Views

Author

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Comments

Links

Crossrefs

Programs

PARI

Formula

Extensions

A268589 a(n) = (2C(3p,p) - 9C(2p,p) + 12) / p^5, where p = prime(n).

A268590 a(n) = (3C(4p,p) - 20C(3p,p) + 54*C(2p,p) - 60) / p^7, where p = prime(n).

A268512 Triangle of coefficients c(n,i), 1<=i<=n, such that for each n>=2, c(n,i) are setwise coprime; and for all primes p>2n-1, the sum of (-1)^ic(n,i)binomial(i*p,p) is divisible by p^(2n-1).