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A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.

%I A147545 #21 Apr 20 2020 00:28:20
%S A147545 3,5,7,11,13,17,23,29,41,47,53,59,83,89,97,107,113,137,167,179,193,
%T A147545 227,233,257,353,359,389,449,467,641,719,769,773,857,929,1097,1283,
%U A147545 1409,1433,1439,1553,1697,1889,2657,2819,2879,3089,3329,3593,3617,3779,5639
%N A147545 Primes of the form p*2^k+1 with k>0 and p=1 or p in this sequence.
%C A147545 This sequence starts like A074781 but grows much faster. Observe that there can be large differences between consecutive terms. Can it be shown that there is always such a prime between consecutive powers of 2? Or that this sequence is infinite? By theorem 1 of the Noe paper, this sequence is a subsequence of A135832, primes in Section I of the phi iteration.
%C A147545 From _Antti Karttunen_, Apr 19 2020: (Start)
%C A147545 Sequence can be considered as a generalization of Fermat primes, A019434, which is a subsequence of this sequence.
%C A147545 All terms with binary weight k (A000120, at least 2 for these terms) can be found as a subset of primes found on the row k-1 of array A334100. E.g. primes with weight 2 are Fermat primes (A019434), those with weight 3 are A334092 (which doesn't contain any other primes), those with weight 4 are in A334093 (among also other kind of primes), those with weights 5, 6, 7 are included as (proper) subsets in A334094, A334095 and A334096 respectively. (End)
%H A147545 T. D. Noe, <a href="/A147545/b147545.txt">Table of n, a(n) for n=1..2000</a>
%H A147545 T. D. Noe, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Noe/noe080107.html">Primes in classes of the iterated totient function</a>, J. Integer Sequences, 11 (2008), Article 08.1.2.
%F A147545 A329697(a(n)) = A000120(a(n)) - 1. - _Antti Karttunen_, Apr 19 2020
%t A147545 nn=2^13; t={1}; i=1; While[q=t[[i]]; k=1; While[p=1+q*2^k; p<nn, If[PrimeQ[p], AppendTo[t,p]]; k++ ]; i<Length[t], i++ ]; t=Rest[Sort[t]]
%o A147545 (PARI)
%o A147545 A000265(n) = (n>>valuation(n,2));
%o A147545 isA147454(n) = ((n>2)&&isprime(n)&&((1==(n=A000265(n-1)))||isA147454(n))); \\ _Antti Karttunen_, Apr 19 2020
%Y A147545 Cf. A000265, A329697, A334100.
%Y A147545 Subsequence of A074781, and of A135832.
%Y A147545 Subsequences: A019434, A334092 (including A039687, A050526, A300407).
%K A147545 nonn
%O A147545 1,1
%A A147545 _T. D. Noe_, Nov 07 2008