A332078 Primes p = k*2^m + 1 such that k + 2^m is not prime, where k and m are the odd part and 2-valuation, respectively, of p-1.

47, 67, 97, 107, 127, 137, 151, 167, 179, 181, 227, 239, 263, 283, 293, 307, 347, 349, 367, 431, 439, 457, 461, 467, 487, 491, 503, 547, 557, 571, 587, 599, 607, 617, 641, 643, 647, 661, 683, 719, 727, 733, 739, 751, 769, 787, 797, 811, 821, 823, 827, 853, 857, 887, 907

Offset: 1

Author

M. F. Hasler, Aug 13 2020

Keywords

• nonn

Comments

It appears that the sequence of odd numbers k*2^m+1 such that k + 2^m is prime (A332075) mainly consists of the primes. This sequence lists the "exceptions": the complement of A332075 within the primes. (The exceptions become more frequent as the numbers grow, the asymptotic density of this subset within the primes might well approach one. See also A332079.)

These are primes of the form p = (w-2^m)*2^m + 1, where w is an odd composite number and 1 < 2^m < w. There are infinitely many primes of this form, because all primes p > 7 such that p == 7 (mod 20) are in this sequence. - Thomas Ordowski, Aug 13 2020

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Thomas Ordowski, Problem, post to the SeqFan list, August 2020.

Crossrefs

Cf. A000040 (primes), A000265 (odd part), A007814 (2-valuation), A332075.

The terms A141882 > 7 are an infinite subsequence. - Thomas Ordowski, Aug 13 2020

Programs

Maple

filter:= proc(p) local k,m;
   if not isprime(p) then return false fi;
   m:= padic:-ordp(p-1,2);
   k:= (p-1)/2^m;
   not isprime(k+2^m);
end proc:
select(filter, [seq(i,i=3..1000,2)]); # Robert Israel, Sep 14 2020

Mathematica

Select[Range[1000], PrimeQ[#] && !PrimeQ[(m = 2^IntegerExponent[# - 1, 2]) + (# - 1)/m] &] (* Amiram Eldar, Aug 14 2020 *)

PARI

(A332078_upto(N)=[p|p<-primes([1,N]),!is_A332075(p)])(1000)