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A046067 Smallest m such that (2n-1)2^m+1 is prime, or -1 if no such value exists. 15


%S 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,

%T 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,

%U 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

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

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

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

%C 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.

%C Jaeschke shows that every positive integer appears infinitely often. - _Jeppe Stig Nielsen_, Jul 06 2020

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

%H T. D. Noe, <a href="/A046067/b046067.txt">Table of n, a(n) for n = 1..5000</a> (with help from the Sierpiński problem website; typo in a(3707)=1 corrected by Jeppe Stig Nielsen)

%H Ray Ballinger and Wilfrid Keller, <a href="http://www.prothsearch.com/sierp.html">Sierpiński Problem</a>

%H John R. Cowles and Ruben Gamboa, <a href="http://arxiv.org/abs/1110.4671">Verifying Sierpiński and Riesel Numbers in ACL2</a>, arXiv preprint arXiv:1110.4671 [cs.DM], 2011.

%H G. Jaeschke, <a href="http://www.jstor.org/stable/2007382">On the Smallest k Such that All k*2^N + 1 are Composite</a>, Mathematics of Computation, Vol. 40, No. 161 (Jan., 1983), pp. 381-384.

%H Seventeen or Bust, <a href="http://www.seventeenorbust.com">A Distributed Attack on the Sierpiński Problem</a>

%H W. Sierpiński, <a href="http://www.digizeitschriften.de/dms/img/?PPN=PPN378850199_0015&amp;DMDID=dmdlog24">Sur un problème concernant les nombres k*2^n+1</a>, Elem. d. Math. 15, pp. 73-74, 1960.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RieselNumber.html">Riesel Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/SierpinskiNumberoftheSecondKind.html">Sierpiński Number of the Second Kind.</a>

%t 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 *)

%Y Cf. A046068.

%Y Bisection of A040076. Cf. A033809.

%Y Cf. A057192, A057025.

%K sign

%O 1,4

%A _Eric W. Weisstein_

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Last modified July 29 09:41 EDT 2021. Contains 346344 sequences. (Running on oeis4.)