cp's OEIS Frontend

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.

%I A332078 #33 Dec 23 2024 14:53:46
%S A332078 47,67,97,107,127,137,151,167,179,181,227,239,263,283,293,307,347,349,
%T A332078 367,431,439,457,461,467,487,491,503,547,557,571,587,599,607,617,641,
%U A332078 643,647,661,683,719,727,733,739,751,769,787,797,811,821,823,827,853,857,887,907
%N 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.
%C A332078 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.)
%C A332078 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
%H A332078 Robert Israel, <a href="/A332078/b332078.txt">Table of n, a(n) for n = 1..10000</a>
%H A332078 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, August 2020.
%p A332078 filter:= proc(p) local k,m;
%p A332078    if not isprime(p) then return false fi;
%p A332078    m:= padic:-ordp(p-1,2);
%p A332078    k:= (p-1)/2^m;
%p A332078    not isprime(k+2^m);
%p A332078 end proc:
%p A332078 select(filter, [seq(i,i=3..1000,2)]); # _Robert Israel_, Sep 14 2020
%t A332078 Select[Range[1000], PrimeQ[#] && !PrimeQ[(m = 2^IntegerExponent[# - 1, 2]) + (# - 1)/m] &] (* _Amiram Eldar_, Aug 14 2020 *)
%o A332078 (PARI) (A332078_upto(N)=[p|p<-primes([1,N]),!is_A332075(p)])(1000)
%Y A332078 Cf. A000040 (primes), A000265 (odd part), A007814 (2-valuation), A332075.
%Y A332078 The terms A141882 > 7 are an infinite subsequence. - _Thomas Ordowski_, Aug 13 2020
%K A332078 nonn
%O A332078 1,1
%A A332078 _M. F. Hasler_, Aug 13 2020