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%I #38 Sep 08 2022 08:46:26
%S 19,31,37,43,61,67,73,79,103,109,127,139,157,163,181,199,223,229,241,
%T 271,277,283,307,313,337,349,367,373,379,397,409,433,439,457,487,499,
%U 523,541,577,607,613,619,643,673,709,733,739,757,787,811,823,829,853,877,907,919
%N Primes p such that (p-1)/gpf(p-1) = 2^q * 3^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.
%C Paul Erdős asked if there are infinitely many primes p such that (p-1)/A006530(p-1) = 2^k or = 2^q*3^r (see Richard K. Guy reference).
%C It is not known if this sequence is infinite.
%C Proposition: if prime p is a term, then p is of the form 6*m+1 (A002476).
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.
%H P. Erdős and C. Pomerance, <a href="http://www.renyi.hu/~p_erdos/1978-29.pdf">On the largest prime factors of n and n+1</a>, Aequationes Math. 17 (1978), pp. 311-321.
%e 31 is prime, 30/5 = 6 = 2*3 hence 31 is a term.
%e 37 is prime, 36/3 = 12 = 2^2*3 hence 37 is a term.
%e 127 is prime, 126/7 = 18 = 2*3^2 hence 127 is a term.
%p alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
%p is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 3}:
%p select(is_a, [$3..919]); # _Peter Luschny_, Dec 13 2020
%t q[n_] := PrimeQ[n] && Module[{f = FactorInteger[n - 1]}, (Length[f] == 2 && f[[2, 1]] == 3 && f[[2, 2]] > 1) || (Length[f] == 3 && f[[2, 1]] == 3 && f[[3, 2]] == 1)]; Select[Range[1000], q] (* _Amiram Eldar_, Dec 09 2020 *)
%o (Magma) s:=func<n|Max(PrimeDivisors(n))>; [p:p in PrimesInInterval(3,1000)|PrimeDivisors(a) eq [2,3] where a is (p-1) div s(p-1)]; // _Marius A. Burtea_, Dec 09 2020
%Y Cf. A006093, A006530, A033845, A052126, A339464.
%Y Cf. A074781 (ratio=2^k), A339466 (ratio <> 2^k and <> 2^q*3^r).
%Y Subsequence of A002476.
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 09 2020
%E More terms from _Marius A. Burtea_, Dec 09 2020
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