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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.
3
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
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: A154844 A383153 A351089 * A325613 A305054 A375148
KEYWORD
nonn
AUTHOR
T. D. Noe, Sep 26 2007
EXTENSIONS
Edited by Peter Munn, Sep 29 2024
STATUS
approved