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

 

Logo

The OEIS is looking to hire part-time people to help edit core sequences, upload scanned documents, process citations, fix broken links, etc. - Neil Sloane, njasloane@gmail.com

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). 27
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.

McIntosh, R. J. (1995), On the converse of Wolstenholme's theorem, Acta Arithmetica 71 (4): 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) = ((binomial (j*k*prime(n), j*prime(n)) - binomial(k*j, j))/(k*prime(n)^3) for k=2, j=1 and n>=3 (conjectured). - 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.

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 May 29 22:45 EDT 2017. Contains 287257 sequences.