A270427 Numbers k such that k*floor(2^k/k) + 1 is prime.

1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 13, 14, 16, 17, 19, 22, 31, 39, 61, 76, 89, 94, 102, 107, 122, 127, 130, 338, 521, 607, 639, 694, 1279, 1352, 1593, 1983, 2061, 2203, 2281, 2319, 2410, 2646, 3217, 4253, 4423, 6345, 7707, 9689, 9941, 11213, 12819, 13175, 14114, 14415, 15293, 19937, 21701, 22839, 23209, 32925, 44497

Offset: 1

Author

Thomas Ordowski, Oct 08 2018

Keywords

• nonn

Comments

Numbers j such that 2^j - (2^j mod j) + 1 is prime.

The associated primes are 2^1+1, 2^2+1, 2^3-1, 2^4+1, 2^5-1, 2^6-3, 2^7-1, 2^8+1, 2^10-3, 2^12-3, 2^13-1, 2^14-3, 2^16+1, 2^17-1, 2^19-1, ...

Are there composite numbers h such that 2^h - (2^(h-1) mod h) is prime?

An odd prime p is in the sequence if and only if 2^p - 1 is prime. Also r = 2^t is a term if and only if 2^r + 1 is an odd prime. So these numbers give all Mersenne primes > 3 and all Fermat primes. Besides, they probably give infinitely many other primes; for example, all primes of the form 4^p - 3 with p prime: 2*p is in the sequence if and only if p is in A058253.

No Fermat pseudoprimes (odd and even) to base 2 in the sequence.

It seems that there are no odd prime powers p^s with s > 1 in the sequence.

Crossrefs

Cf. A000043, A019434, A058253, A128092.

Programs

Mathematica

Select[Range[7000], PrimeQ[#*Floor[2^#/#] + 1] &] (* G. C. Greubel, Oct 09 2018 *)

PARI

for(n=1, 1000, if(isprime(n * floor(2^n/n) + 1), print1(n,", "))) \\ Amiram Eldar, Oct 09 2018

Extensions

Three missing terms supplemented by Amiram Eldar, Oct 09 2018

a(47)-a(49) added by G. C. Greubel, Oct 09 2018

a(50)-a(61) added by Amiram Eldar, Oct 09 2018