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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 (list; graph; refs; listen; history; text; internal format)
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 proposes 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

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

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. A006093, A006530, A052126, A339464.

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 A195270

Adjacent sequences:  A339463 A339464 A339465 * A339467 A339468 A339469

KEYWORD

nonn

AUTHOR

Bernard Schott, Dec 10 2020

EXTENSIONS

More terms from Amiram Eldar, Dec 11 2020

STATUS

approved

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Last modified August 5 15:11 EDT 2021. Contains 346471 sequences. (Running on oeis4.)