A135516 a(0)=1; a(n) = (Product_{i=1..n} prime(i)^2) - 1, where prime(i) is the i-th prime.

1, 3, 35, 899, 44099, 5336099, 901800899, 260620460099, 94083986096099, 49770428644836899, 41856930490307832899, 40224510201185827416899, 55067354465423397733736099, 92568222856376731590410384099

Offset: 0

Author

Ctibor O. Zizka, Feb 19 2008

Keywords

• easy
• nonn

Comments

Sequence can be generalized: a(0)=1; a(n) = (Product_{i=1..n} prime(i)^r) - 1, where prime(i) is the i-th prime.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..99 [Offset shifted by _Georg Fischer_, Jun 18 2021]
• A. Adelberg, S. Hong and W. Ren, Bounds on divided universal Bernoulli numbers and universal Kummer congruences, Proc. Amer. Math. Soc., Volume 136, Number 1, 2008, Pages 61-71,
• Alexei A. Panchishkin, Generalized Kummer congruences and p-adic families of motives, arXiv:math/9503218 [math.NT], 1995.

Crossrefs

Cf. A057588, A057705, A002110.

Programs

Maple

A002110 := proc(n) mul(ithprime(i),i=1..n) ; end:
A135516 := proc(n) if n =0 then 1; else (A002110(n)+1)*(A002110(n)-1) ; fi ; end: seq(A135516(n),n=0..20) ; # R. J. Mathar, Feb 28 2008

Mathematica

Join[{1},Rest[#-1&/@FoldList[Times,1,Prime[Range[15]]^2]]] (* Harvey P. Dale, Oct 02 2011 *)
Join[{1}, Table[Product[Prime[i]^(2), {i,1,n}] - 1, {n,1,15}]] (* G. C. Greubel, Oct 17 2016 *)

PARI

a(n) = if(n==0, 1, prod(k=1, n, prime(k)^2) - 1); \\ Michel Marcus, Oct 17 2016

Formula

a(n) = A061742(n-1)-1 = (A002110(n)+1)*(A002110(n)-1) for n>1. - R. J. Mathar, Feb 28 2008

Extensions

Offset corrected by Georg Fischer, Jun 18 2021