login
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.
4
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
1,1
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
LINKS
Thomas Ordowski, Problem, post to the SeqFan list, Aug 11 2020.
EXAMPLE
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.
MATHEMATICA
Select[Range[3, 251, 2], PrimeQ[(m = 2^IntegerExponent[# - 1, 2]) + (# - 1)/m] &] (* Amiram Eldar, Aug 14 2020 *)
PROG
(PARI) select( is_A332075(n)=n%2&&n>1&&ispseudoprime((n>>n=valuation(n-1, 2))+2^n), [1..255])
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).
Sequence in context: A118749 A367227 A355821 * A122089 A109860 A100392
KEYWORD
nonn
AUTHOR
M. F. Hasler, Aug 13 2020
STATUS
approved