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Primes of the form p*2^k + 1 for any k and any prime p.
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%I #44 Oct 26 2024 10:29:59

%S 3,5,7,11,13,17,23,29,41,47,53,59,83,89,97,107,113,137,149,167,173,

%T 179,193,227,233,257,263,269,293,317,347,353,359,383,389,449,467,479,

%U 503,509,557,563,569,587,593,641,653,719,769,773,797,809,839,857,863,887

%N Primes of the form p*2^k + 1 for any k and any prime p.

%C From _Bernard Schott_, Dec 14 2020: (Start)

%C Equivalently, primes p such that the ratio (p-1)/gpf(p-1) = 2^k where gpf(m) is the greatest prime factor of m, A006530.

%C Paul Erdős asked if there are infinitely many primes p in this sequence (see R. K. Guy reference). (End)

%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.

%H T. D. Noe, <a href="/A074781/b074781.txt">Table of n, a(n) for n = 1..2000</a>

%H Graeme L. Cohen, <a href="https://eudml.org/doc/210499">On a conjecture of Makowski and Schinzel</a>, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8.

%e 3 = 2*2^0+1 is a term and 2/2 = 1 = 2^0.

%e 7 = 3*2^1+1 is a term and 6/3 = 2 = 2^1.

%e 13 = 3*2^2+1 is a term and 12/3 = 4 = 2^2.

%e 41 = 5*2^3+1 is a term and 40/5 = 8 = 2^3.

%e 113 = 7*2^4+1 is a term and 112/7 = 16 = 2^4.

%p alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):

%p is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:

%p 3, op(select(is_a, [$3..919])); # _Peter Luschny_, Dec 14 2020

%t Select[Range[3, 1000], PrimeQ[#] && !CompositeQ[(# - 1)/2^IntegerExponent[(# - 1), 2]] &] (* _Amiram Eldar_, Dec 28 2018 *)

%Y Cf. A000079, A006093, A006530, A052126, A339464.

%Y Cf. other ratios : A339463, A339465, A339466.

%Y Subsequences: A039687, A051900, A058500 (this sequence without the Fermat primes), A090866, A147545,

%K nonn,changed

%O 1,1

%A Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002