A179289 Smallest index k such that prime(k)*2^n-1 is prime, or zero if there is no prime.

1, 1, 2, 1, 4, 1, 2, 3, 4, 3, 2, 1, 11, 3, 22, 1, 4, 1, 18, 7, 4, 23, 6, 23, 18, 5, 44, 23, 4, 1, 14, 3, 11, 2, 11, 7, 11, 2, 18, 28, 8, 16, 2, 102, 4, 9, 11, 3, 8, 5, 174, 24, 63, 3, 2, 103, 22, 23, 130, 1, 22, 16, 18, 2, 19, 55, 14, 41, 34, 15

Offset: 1

Author

Pierre CAMI, Jul 09 2010

Keywords

• nonn

Comments

Define partial sum S(N)=Sum(n=1...N) of n , T(N)=Sum(n=1...N) of k(n) The ratio T(N)/S(N) --> approx 0.5236 as N --> infinity.

It is conjectured that a(42228) is the first 0 term. This corresponds to the first Riesel number, 509203, which happens to be prime. See A101036. - T. D. Noe, Mar 23 2011

Crossrefs

Cf. A126715.

Programs

Mathematica

sik[n_]:=Module[{c=2^n,k=1},While[!PrimeQ[Prime[k]*c-1],k++];k]; Array[ sik,70] (* The program will NOT identify cases where no prime satisfies the definition.  See the second comment. *) (* Harvey P. Dale, Jan 10 2016 *)

Other

SCRIPT / DIM nn,0 / DIM kk / DIMS st / LABEL loopn / SET nn,nn+1 / IF nn>10000 THEN END / SET kk,0 / LABEL loopk / SET kk,kk+1 / SET st,%d,%d,%d\,;nn;kk;p(kk) / PRP p(kk)*2^nn-1 / IF ISPRIME THEN GOTO loopn / GOTO loopk / / This file is the in.txt file / The command is PFGW -f in.txt / The results are in the file pfgw-prime.log for small n / and in the pfgw.log file for greatest n / Program PFGW from Primeform Group /

PARI

a(n) = {my(k=1); while (!isprime(prime(k)*2^n-1), k++); k;} \\ Michel Marcus, Sep 16 2019

Formula

a(n) = 1 for n = A000043(k) - 1, the Mersenne exponents minus 1. - T. D. Noe, Mar 23 2011