login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual appeal: Please make a donation to keep the OEIS running! Over 6000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A034602 Wolstenholme quotient W_p = (binomial(2p-1,p) - 1)/p^3 for prime p=A000040(n). 28
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

Table of n, a(n) for n=3..18.

R. Aidagulov and M. A. Alekseyev, On p-adic approximation of sums of binomial coefficients. Journal of Mathematical Sciences, 2017. (in press). Also arXiv preprint arXiv:1602.02632, 2016.

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.

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.

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

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified November 18 19:06 EST 2017. Contains 294894 sequences.