login
A339466
Primes p such that gpf((p - 1)/gpf(p - 1)) > 3, where gpf(m) is the greatest prime factor of m, A006530.
6
71, 101, 131, 151, 191, 197, 211, 239, 251, 281, 311, 331, 401, 419, 421, 431, 443, 461, 463, 491, 521, 547, 571, 599, 601, 617, 631, 647, 659, 661, 677, 683, 691, 701, 727, 743, 751, 761, 821, 827, 859, 881, 883, 911, 941, 947, 953, 967, 971, 991, 1013, 1021
OFFSET
1,1
COMMENTS
Paul Erdős asked if there are infinitely many primes p such that (p-1)/gpf(p-1) = 2^k or = 2^q * 3^r (see Richard K. Guy reference). This sequence lists the primes p that do not satisfy these two previous relations.
Replacing in the definition gpf by lpf (A020639) leads to A122259. In fact this sequence is a subsequence of A122259. - Peter Luschny, Dec 13 2020
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), p. 311-321. [alternate link]
EXAMPLE
71 is prime, 70/7 = 10 = 2*5 hence 71 is a term.
101 is prime, 100/5 = 20 = 2^2*5 hence 101 is a term.
151 is prime, 150/5 = 30 = 2*3*5 hence 151 is a term.
The first few quotients obtained are: 10, 20, 10, 30, 10, 28, 30, 14, 50, 40, ...
MAPLE
alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
is_a := n -> isprime(n) and gpf((n-1)/gpf(n-1)) > 3:
select(is_a, [$5..1021]); # Peter Luschny, Dec 13 2020
MATHEMATICA
q[n_] := Module[{f = FactorInteger[n - 1]}, (Length[f] == 1 && f[[1, 1]] == 2) || (Length[f] == 2 && f[[1, 1]] == 2 && f[[2, 2]] == 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[3, 1000], PrimeQ[#] && ! q[#] &] (* Amiram Eldar, Dec 10 2020 *)
PROG
(Magma) s:=func<n|Max(PrimeDivisors(n))>; [p:p in PrimesInInterval(3, 1100)|( not 3 in PrimeDivisors(a) and #PrimeDivisors(a) ge 2) or ( 3 in PrimeDivisors(a) and #PrimeDivisors(a) ge 3) where a is (p-1) div s(p-1)]; // Marius A. Burtea, Dec 10 2020
(PARI) is(n) = {if(!isprime(n) || n==2, return(0)); my(pm1 = n-1, f = factor(pm1)[, 1]); (pm1 /= (f[#f]*1<<valuation(pm1, 2)*3^valuation(pm1, 3)))>1} \\ David A. Corneth, Dec 13 2020
CROSSREFS
Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r), A339463 (ratio=2^q*5^r).
Cf. A122259.
Sequence in context: A288907 A234962 A166252 * A339463 A166576 A369250
KEYWORD
nonn
AUTHOR
Bernard Schott, Dec 10 2020
EXTENSIONS
More terms from Amiram Eldar, Dec 11 2020
STATUS
approved