A046067 Smallest m such that (2n-1)2^m+1 is prime, or -1 if no such value exists.

0, 1, 1, 2, 1, 1, 2, 1, 3, 6, 1, 1, 2, 2, 1, 8, 1, 1, 2, 1, 1, 2, 2, 583, 2, 1, 1, 4, 2, 5, 4, 1, 1, 2, 1, 3, 2, 1, 3, 2, 1, 1, 4, 2, 1, 8, 2, 1, 2, 1, 3, 16, 1, 3, 6, 1, 1, 2, 3, 1, 8, 6, 1, 2, 3, 1, 4, 1, 3, 2, 1, 53, 6, 8, 3, 4, 1, 1, 8, 6, 3, 2, 1, 7, 2, 8, 1, 2, 2, 1, 4, 1, 3, 6, 1, 1, 2, 4, 15, 2

Offset: 1

Author

Eric W. Weisstein

Keywords

• sign

Comments

There exist odd integers 2k-1 such that (2k-1)2^n+1 is always composite.

The smallest known example is 78557. Therefore a(39279) = -1.

For the corresponding primes see A057025(n-1), n >= 1, where a 0 will show up if a(n) = -1. - Wolfdieter Lang, Feb 07 2013.

Jaeschke shows that every positive integer appears infinitely often. - Jeppe Stig Nielsen, Jul 06 2020

References

• Ribenboim, P. The New Book of Prime Number Records. New York: Springer-Verlag, pp. 357-359, 1996.

Links

• T. D. Noe, Table of n, a(n) for n = 1..5000 (with help from the Sierpiński problem website; typo in a(3707)=1 corrected by Jeppe Stig Nielsen)
• Ray Ballinger and Wilfrid Keller, Sierpiński Problem
• John R. Cowles and Ruben Gamboa, Verifying Sierpiński and Riesel Numbers in ACL2, arXiv preprint arXiv:1110.4671 [cs.DM], 2011.
• G. Jaeschke, On the Smallest k Such that All k*2^N + 1 are Composite, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.
• Seventeen or Bust, A Distributed Attack on the Sierpiński Problem
• W. Sierpiński, Sur un problème concernant les nombres k*2^n+1, Elem. d. Math. 15, pp. 73-74, 1960.
• Eric Weisstein's World of Mathematics, Riesel Number.
• Eric Weisstein's World of Mathematics, Sierpiński Number of the Second Kind.

Crossrefs

Cf. A046068.

Bisection of A040076. Cf. A033809.

Cf. A057192, A057025.

Programs

Mathematica

max = 10000 (* this maximum value of m is sufficient up to n = 1000 *); a[n_] := For[m = 1, m <= max, m++, If[PrimeQ[(2n - 1)*2^m + 1], Return[m]]] /. Null -> -1; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Jun 08 2012 *)