A339465 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.

19, 31, 37, 43, 61, 67, 73, 79, 103, 109, 127, 139, 157, 163, 181, 199, 223, 229, 241, 271, 277, 283, 307, 313, 337, 349, 367, 373, 379, 397, 409, 433, 439, 457, 487, 499, 523, 541, 577, 607, 613, 619, 643, 673, 709, 733, 739, 757, 787, 811, 823, 829, 853, 877, 907, 919

Offset: 1

Author

Bernard Schott, Dec 09 2020

Keywords

• nonn

Comments

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).

It is not known if this sequence is infinite.

Proposition: if prime p is a term, then p is of the form 6*m+1 (A002476).

Examples

			31 is prime, 30/5 = 6 = 2*3 hence 31 is a term.
37 is prime, 36/3 = 12 = 2^2*3 hence 37 is a term.
127 is prime, 126/7 = 18 = 2*3^2 hence 127 is a term.
		

References

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

Links

• P. Erdős and C. Pomerance, On the largest prime factors of n and n+1, Aequationes Math. 17 (1978), pp. 311-321.

Crossrefs

Cf. A006093, A006530, A033845, A052126, A339464.

Cf. A074781 (ratio=2^k), A339466 (ratio <> 2^k and <> 2^q*3^r).

Subsequence of A002476.

Programs

Magma

s:=func; [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

Maple

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 3}:
select(is_a, [$3..919]); # Peter Luschny, Dec 13 2020

Mathematica

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 *)

Extensions

More terms from Marius A. Burtea, Dec 09 2020