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

%I #71 Oct 26 2018 10:32:48

%S 1,2,3,4,5,6,7,8,10,12,13,14,16,17,19,22,31,39,61,76,89,94,102,107,

%T 122,127,130,338,521,607,639,694,1279,1352,1593,1983,2061,2203,2281,

%U 2319,2410,2646,3217,4253,4423,6345,7707,9689,9941,11213,12819,13175,14114,14415,15293,19937,21701,22839,23209,32925,44497

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

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

%C 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, ...

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

%C 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.

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

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

%t Select[Range[7000], PrimeQ[#*Floor[2^#/#] + 1] &] (* _G. C. Greubel_, Oct 09 2018 *)

%o (PARI) for(n=1, 1000, if(isprime(n * floor(2^n/n) + 1), print1(n,", "))) \\ _Amiram Eldar_, Oct 09 2018

%Y Cf. A000043, A019434, A058253, A128092.

%K nonn

%O 1,2

%A _Thomas Ordowski_, Oct 08 2018

%E Three missing terms supplemented by _Amiram Eldar_, Oct 09 2018

%E a(47)-a(49) added by _G. C. Greubel_, Oct 09 2018

%E a(50)-a(61) added by _Amiram Eldar_, Oct 09 2018

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