A074781 Primes of the form p*2^k + 1 for any k and any prime p.

3, 5, 7, 11, 13, 17, 23, 29, 41, 47, 53, 59, 83, 89, 97, 107, 113, 137, 149, 167, 173, 179, 193, 227, 233, 257, 263, 269, 293, 317, 347, 353, 359, 383, 389, 449, 467, 479, 503, 509, 557, 563, 569, 587, 593, 641, 653, 719, 769, 773, 797, 809, 839, 857, 863, 887

Offset

1,1

Comments

From Bernard Schott, Dec 14 2020: (Start)

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.

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

References

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

Links

T. D. Noe, Table of n, a(n) for n = 1..2000

Graeme L. Cohen, On a conjecture of Makowski and Schinzel, Colloquium Mathematicae, Vol. 74, No. 1 (1997), pp. 1-8.

Example

3 = 2*2^0+1 is a term and 2/2 = 1 = 2^0.
7 = 3*2^1+1 is a term and 6/3 = 2 = 2^1.
13 = 3*2^2+1 is a term and 12/3 = 4 = 2^2.
41 = 5*2^3+1 is a term and 40/5 = 8 = 2^3.
113 = 7*2^4+1 is a term and 112/7 = 16 = 2^4.

Maple

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2}:
3, op(select(is_a, [$3..919])); # Peter Luschny, Dec 14 2020

Mathematica

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

Crossrefs

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

Cf. other ratios : A339463, A339465, A339466.

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

Sequence in context: A075794 A192864 A135832 * A147545 A083668 A176116

Adjacent sequences: A074778 A074779 A074780 * A074782 A074783 A074784

Keyword

nonn

Author

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

Status

approved