A339463 Primes p such that (p-1)/gpf(p-1) = 2^q * 5^r with q, r >= 1, where gpf(m) is the greatest prime factor of m, A006530.

71, 101, 131, 191, 251, 281, 311, 401, 431, 461, 521, 701, 761, 821, 881, 941, 971, 1031, 1061, 1091, 1151, 1181, 1301, 1361, 1451, 1481, 1511, 1571, 1601, 1721, 1811, 1901, 1931, 2081, 2111, 2141, 2351, 2411, 2441, 2621, 2711, 2741, 2801, 3041, 3251, 3371

Offset: 1

Author

Bernard Schott, Dec 13 2020

Keywords

• nonn

Comments

These primes that are all congruent to 11 (mod 30) form a subsequence of A132232. The first terms of A132232 that are not terms here are 11, 41, 491, ... (see examples)

Examples

			41 is prime, 40/5 = 8 = 2^3, hence 41 is not a term.
101 is prime, 100/5 = 20 = 2^2 * 5, hence 101 is a term.
491 is prime, 490/7 = 70 = 2 * 5 * 7, hence 491 is not a term.
521 is prime, 520/13 = 40 = 2^3 * 5, hence 521 is a term.
		

References

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

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A006093 (prime(n)-1), A006530, A052126, A339464.

Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r).

Subsequence of A132232 and of A339466.

Programs

Maple

alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and pf((n-1)/gpf(n-1)) = {2, 5}:
select(is_a, [$5..3371]); # Peter Luschny, Dec 13 2020

Mathematica

q[n_] := Divisible[n, 10] && ((PrimeQ[(r = n/2^IntegerExponent[n, 2]/5^(e = IntegerExponent[n, 5]))] && r > 5) || (r == 1 && e > 1)); Select[Range[3500], PrimeQ[#] && q[# - 1] &] (* Amiram Eldar, Dec 13 2020 *)