A217772 -id:A217772 - 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

A223886 Numbers (binomial(jkprime(n), jprime(n)) - binomial(kj, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.

Original entry on oeis.org

871695, 106388178385, 23847838715080655, 2591856748839247419391825095, 1049841259371423735816549330164685, 216822871259048720341882553570648156557191421

Offset: 2

Alexander R. Povolotsky, Mar 28 2013

nonn

This sequence (together with already present in the OEIS A034602 and A217772) is based on Gary Detlefs' conjecture, which he disclosed to me in a private communication on 3/29/13 and recently he gave me permission to make it public. Specifically he wrote to me the following: "I have a conjecture which is broader than the one I submitted, having to do with binomial(k*n,n) mod n^3. It appears that binomial(j*k*n,j*n) mod n^3 will be binomial(k*j,j) for n sufficiently large."
In effect above conjecture further extends Wolstenholme's and Ljunggren's ideas and could also be expressed as follows: starting with some specific (for any given unchanged values of integers k>0 and j>0) sufficiently large value of n=N and further on for n>N it is true that (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/k/(prime(n))^3 = m(j, k, n ), where m(j, k, n ) are integer values.
Note that the values of A034602 are replicated by above general formula for k=2, j=1 and n>=3 and the values of A217772 are replicated by the same formula for k=3, j=1 and n>=2.

R. Mestrovic, Wolstenholme's theorem: Its Generalizations and Extensions in the last hundred and fifty years (1862-2011), arXiv preprint arXiv:1111.3057 [math.NT], 2011.

Cf. A034602, A217772.

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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A223886 Numbers (binomial(jkprime(n), jprime(n)) - binomial(kj, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.

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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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A223886 Numbers (binomial(j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.

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A223886 Numbers (binomial(jkprime(n), jprime(n)) - binomial(kj, j))/(k*prime(n)^3) for k=4, j=3 and n>=2.