

A270427


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


2



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
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OFFSET

1,2


COMMENTS

Numbers j such that 2^j  (2^j mod j) + 1 is prime.
The associated primes are 2^1+1, 2^2+1, 2^31, 2^4+1, 2^51, 2^63, 2^71, 2^8+1, 2^103, 2^123, 2^131, 2^143, 2^16+1, 2^171, 2^191, ...
Are there composite numbers h such that 2^h  (2^(h1) 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.


LINKS

Table of n, a(n) for n=1..61.


MATHEMATICA

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


PROG

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


CROSSREFS

Cf. A000043, A019434, A058253, A128092.
Sequence in context: A039164 A263977 A335040 * A270189 A257672 A285314
Adjacent sequences: A270424 A270425 A270426 * A270428 A270429 A270430


KEYWORD

nonn


AUTHOR

Thomas Ordowski, Oct 08 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


STATUS

approved



