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
KEYWORD
nonn
AUTHOR
Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Sep 07 2002
STATUS
approved