A332079 Number of primes between 2^n and the least prime p > 2^n in A332075, i.e., such that k + 2^m is prime, where k and m are the odd part and 2-valuation, respectively, of p-1.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 1, 8, 1, 9, 7, 0, 0, 7, 5, 1, 2, 4, 9, 1, 7, 8, 6, 11, 0, 4, 0, 1, 1, 0, 0, 10, 17, 3, 0, 8, 0, 10, 20, 3, 23, 15, 3, 20, 13, 7, 36, 17, 15, 4, 4, 0, 9, 15, 10, 21, 8, 22, 36, 6, 13, 2, 7, 36, 14, 10, 9, 4, 0, 44, 10, 8, 27, 5, 1, 0, 2, 22, 3, 2, 33, 20, 21, 19, 12, 12, 5

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 primes, and many primes are in this sequence. This sequence attempts to measure in how far this remains true for large numbers.

Links

• T. Ordowski, Problem, post to the SeqFan list, Aug 11 2020.

Crossrefs

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

Programs

Mathematica

a[n_] := Module[{count = 0, p = NextPrime[2^n]}, While[!PrimeQ[(m = 2^IntegerExponent[p - 1, 2]) + (p - 1)/m], count++; p = NextPrime[p]]; count]; s = Array[a, 100] (* Amiram Eldar, Aug 14 2020 *)

PARI

apply( {A332079(n,c=0)=forprime(p=2^n,,is_A332075(p)&&return(c);c++)}, [1..99])