A332075 Odd numbers 2n+1 such that k + 2^m is prime, where k and m are the odd part and 2-valuation, respectively, of 2n.

3, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 35, 37, 41, 43, 49, 53, 55, 59, 61, 71, 73, 77, 79, 83, 89, 91, 101, 103, 109, 113, 115, 119, 121, 131, 133, 139, 143, 149, 155, 157, 161, 163, 169, 173, 175, 185, 191, 193, 197, 199, 203, 209, 211, 215, 221, 223, 229, 233, 241, 251

Offset: 1

Author

M. F. Hasler, Aug 13 2020

Keywords

• nonn

Comments

It appears that about 1/log_10(N) of the odd numbers below 2N are in this sequence: for n < 10^k with k = (1, 2, 3, 4, 5, 6), there are (7, 51, 364, 2675, 20668, 167185) numbers as defined in NAME.

We observe that the sequence mainly consists of the odd primes but some of them are missing (47, 67, 97, 107, 127, 137, 151, 167, ...) and there are some composite terms {25, 35, 49, 55, 77, 91, ...}.

The frequency of primes in this sequence remains high: the least prime > 10^99 with this property is only 10^99 + 2191. See A332078 for primes not in this sequence.

These are numbers of the form (p-2^m)*2^m + 1, where p is an odd prime and 1 < 2^m < p, so there are infinitely many such numbers. Problem: are there infinitely many primes of this form? All the numbers A016945 > 3 are not in this sequence. - Thomas Ordowski, Aug 13 2020

Otherwise said, 3 is the only term divisible by 3. - M. F. Hasler, Aug 14 2020

Examples

			For the smallest odd number 1 = 2*0 + 1, 2n = 2*0 has no well defined 2-adic valuation, so it is omitted here.
For the next odd number 3 = 2*1 + 1, 2n = 2 has odd part (A000265) and 2-adic valuation (A007814) both equal to k = m = 1, and k + 2^m = 3 is prime.
		

Links

• Michel Marcus, Table of n, a(n) for n = 1..6000
• Thomas Ordowski, Problem, post to the SeqFan list, Aug 11 2020.

Crossrefs

Cf. A000040 (primes), A000265 (odd part), A007814 (2-valuation), A332078 (primes that are not in this sequence), A332079 (number of primes between 2^n and the next larger prime in this sequence).

Programs

Mathematica

Select[Range[3, 251, 2], PrimeQ[(m = 2^IntegerExponent[# - 1, 2]) + (# - 1)/m] &] (* Amiram Eldar, Aug 14 2020 *)

PARI

select( is_A332075(n)=n%2&&n>1&&ispseudoprime((n>>n=valuation(n-1,2))+2^n), [1..255])