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A034602 Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n). 29
1, 5, 265, 2367, 237493, 2576561, 338350897, 616410400171, 7811559753873, 17236200860123055, 3081677433937346539, 41741941495866750557, 7829195555633964779233, 21066131970056662377432067, 59296957594629000880904587621, 844326030443651782154010715715 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

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

REFERENCES

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

LINKS

Robert Israel, Table of n, a(n) for n = 3..263

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 arXiv:1602.02632

R. J. McIntosh, On the converse of Wolstenholme's theorem, Acta Arithmetica 71:4 (1995), 381-389.

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

J. 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.

FORMULA

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

EXAMPLE

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

MAPLE

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

MATHEMATICA

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

PROG

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

CROSSREFS

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

Cf. A268512, A268589, A268590.

Sequence in context: A159954 A079681 A086656 * A175180 A238799 A113257

Adjacent sequences:  A034599 A034600 A034601 * A034603 A034604 A034605

KEYWORD

nonn

AUTHOR

N. J. A. Sloane.

EXTENSIONS

Edited by Max Alekseyev, May 14 2010

More terms from Vincenzo Librandi, Nov 23 2015

STATUS

approved

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Last modified January 17 19:58 EST 2019. Contains 319251 sequences. (Running on oeis4.)