A133831 Least positive number k != n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

2, 1, 1, 1, 2, 1, 1, 9, 3, 3, 2, 1, 4, 5, 1, 1, 11, 1, 6, 5, 4, 7, 3, 9, 27, 17, 15, 1, 15, 1, 6, 458465, 4, 9, 14, 13, 3, 11, 25, 57, 6, 7, 46, 17, 7, 15, 2, 1009, 30, 23, 6, 21, 2, 33, 1, 1265, 3, 69, 14, 5, 6, 21, 19, 2241, 30, 3, 1, 5, 34, 19, 26, 17, 19, 17, 5, 33, 15, 23, 27

Offset

1,1

Comments

Does such k exist (so that a(n) is nonzero) for all n? These binary trinomials can also be written as f*2^n+1, where f=2^m+1 for some m, which is reminiscent of the Sierpinski problem (see A076336). Hence if there are no Sierpinski numbers of the form 2^m+1, then a(n) is nonzero for all n.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime. If a(256) is nonzero, it is greater than 10^6.

Links

T. D. Noe, Table of n, a(n) for n = 1..255

Henri Lifchitz and Renaud Lifchitz (Editors), Search for 2^n+2^m+1, PRP Top Records.

Mathematica

mx=4000; Table[s=1+2^n; k=1; While[k==n || (k<mx && !PrimeQ[s+2^k]), k++ ]; If[k==mx, 0, k], {n, 100}]

Crossrefs

Cf. A057732, A059242, A057196, A057200, A081091 (various forms of prime binary trinomials).

Closely related problems: A040076 (see also A076336), A067760, A133830 (k < n), A133832 (k > n).

Cf. A095056.

Sequence in context: A290084 A154844 A351089 * A325613 A305054 A375148

Adjacent sequences: A133828 A133829 A133830 * A133832 A133833 A133834

Keyword

nonn

Author

T. D. Noe, Sep 26 2007

Extensions

Edited by Peter Munn, Sep 29 2024

Status

approved