cp's OEIS Frontend

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.

%I A332075 #41 Dec 23 2024 14:53:46
%S A332075 3,5,7,11,13,17,19,23,25,29,31,35,37,41,43,49,53,55,59,61,71,73,77,79,
%T A332075 83,89,91,101,103,109,113,115,119,121,131,133,139,143,149,155,157,161,
%U A332075 163,169,173,175,185,191,193,197,199,203,209,211,215,221,223,229,233,241,251
%N 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.
%C A332075 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.
%C A332075 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, ...}.
%C A332075 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.
%C A332075 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
%C A332075 Otherwise said, 3 is the only term divisible by 3. - _M. F. Hasler_, Aug 14 2020
%H A332075 Michel Marcus, <a href="/A332075/b332075.txt">Table of n, a(n) for n = 1..6000</a>
%H A332075 Thomas Ordowski, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2020-August/020922.html">Problem</a>, post to the SeqFan list, Aug 11 2020.
%e A332075 For the smallest odd number 1 = 2*0 + 1, 2n = 2*0 has no well defined 2-adic valuation, so it is omitted here.
%e A332075 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.
%t A332075 Select[Range[3, 251, 2], PrimeQ[(m = 2^IntegerExponent[# - 1, 2]) + (# - 1)/m] &] (* _Amiram Eldar_, Aug 14 2020 *)
%o A332075 (PARI) select( is_A332075(n)=n%2&&n>1&&ispseudoprime((n>>n=valuation(n-1,2))+2^n), [1..255])
%Y A332075 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).
%K A332075 nonn
%O A332075 1,1
%A A332075 _M. F. Hasler_, Aug 13 2020