A133830 -id:A133830 - cp's OEIS frontend

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

T. D. Noe, Sep 26 2007

nonn

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.

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.

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.

mx=4000; Table[s=1+2^n; k=1; While[k==n || (k

Edited by Peter Munn, Sep 29 2024

A133832 Least number k > n such that the binary trinomial 1 + 2^n + 2^k is prime, or 0 if there is no such k.

Original entry on oeis.org

2, 3, 5, 13, 6, 7, 9, 9, 18, 19, 14, 13, 15, 17, 17, 81, 20, 19, 30, 33, 26, 27, 38, 81, 27, 35, 31, 33, 35, 31, 42, 458465, 36, 45, 47, 37, 67, 53, 41, 57, 42, 45, 46, 69, 54, 57, 53, 1009, 100, 119, 55, 73, 83, 67, 57, 1265, 74, 69, 66, 113, 75, 101, 66, 2241, 68, 67, 70

Offset: 1

T. D. Noe, Sep 26 2007

nonn

Conjecture: 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). The conjecture is equivalent to no Sierpinski numbers of the form 2^m+1 existing.

The PFGW program was used to find a(32), which produces a 138012-digit probable prime.

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.

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

Cf. A095056, A133830 (k < n equivalent), A133831.

mx=4000; Table[s=1+2^n; k=n+1; While[k

Edited by Peter Munn, Sep 29 2024

cp's OEIS Frontend

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.

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