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

1,1

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.

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 *)

Prog

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

Crossrefs

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

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

Sequence in context: A099591 A046048 A163390 * A023303 A348421 A106874

Adjacent sequences: A332075 A332076 A332077 * A332079 A332080 A332081

Keyword

nonn,changed

Author

M. F. Hasler, Aug 13 2020

Status

approved