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

1,1

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).

References

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

Links

Table of n, a(n) for n=1..56.

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

Example

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.

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 *)

Prog

(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

Crossrefs

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

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

Subsequence of A002476.

Sequence in context: A125148 A214798 A040066 * A279192 A159043 A040088

Adjacent sequences: A339462 A339463 A339464 * A339466 A339467 A339468

Keyword

nonn

Author

Bernard Schott, Dec 09 2020

Extensions

More terms from Marius A. Burtea, Dec 09 2020

Status

approved