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

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 167, 179, 193, 227, 233, 257, 353, 359, 389, 449, 467, 641, 719, 769, 773, 857, 929, 1097, 1283, 1409, 1433, 1439, 1553, 1697, 1889, 2657, 2819, 2879, 3089, 3329, 3593, 3617, 3779, 5639

Offset: 1

Author

T. D. Noe, Nov 07 2008

Keywords

• nonn

Comments

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.

From Antti Karttunen, Apr 19 2020: (Start)

Sequence can be considered as a generalization of Fermat primes, A019434, which is a subsequence of this sequence.

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)

Links

• T. D. Noe, Table of n, a(n) for n=1..2000
• T. D. Noe, Primes in classes of the iterated totient function, J. Integer Sequences, 11 (2008), Article 08.1.2.

Crossrefs

Cf. A000265, A329697, A334100.

Subsequence of A074781, and of A135832.

Subsequences: A019434, A334092 (including A039687, A050526, A300407).

Programs

Mathematica

nn=2^13; t={1}; i=1; While[q=t[[i]]; k=1; While[p=1+q*2^k; p
				

PARI

A000265(n) = (n>>valuation(n,2));
isA147454(n) = ((n>2)&&isprime(n)&&((1==(n=A000265(n-1)))||isA147454(n))); \\ Antti Karttunen, Apr 19 2020

Formula

A329697(a(n)) = A000120(a(n)) - 1. - Antti Karttunen, Apr 19 2020