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%I #49 Mar 31 2023 09:18:32
%S 71,101,131,151,191,197,211,239,251,281,311,331,401,419,421,431,443,
%T 461,463,491,521,547,571,599,601,617,631,647,659,661,677,683,691,701,
%U 727,743,751,761,821,827,859,881,883,911,941,947,953,967,971,991,1013,1021
%N Primes p such that gpf((p - 1)/gpf(p - 1)) > 3, where gpf(m) is the greatest prime factor of m, A006530.
%C 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.
%C 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
%D Richard K. Guy, Unsolved Problems in Number Theory, 3rd Edition, Springer, 2004, Section B46, p. 154.
%H Robert Israel, <a href="/A339466/b339466.txt">Table of n, a(n) for n = 1..10000</a>
%H P. Erdős and C. Pomerance, <a href="http://www.math.dartmouth.edu/~carlp/PDF/paper17.pdf">On the largest prime factors of n and n+1</a>, Aequationes Math. 17 (1978), p. 311-321. [<a href="http://www.renyi.hu/~p_erdos/1978-29.pdf">alternate link</a>]
%e 71 is prime, 70/7 = 10 = 2*5 hence 71 is a term.
%e 101 is prime, 100/5 = 20 = 2^2*5 hence 101 is a term.
%e 151 is prime, 150/5 = 30 = 2*3*5 hence 151 is a term.
%e The first few quotients obtained are: 10, 20, 10, 30, 10, 28, 30, 14, 50, 40, ...
%p alias(pf = NumberTheory:-PrimeFactors): gpf := n -> max(pf(n)):
%p is_a := n -> isprime(n) and gpf((n-1)/gpf(n-1)) > 3:
%p select(is_a, [$5..1021]); # _Peter Luschny_, Dec 13 2020
%t 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 *)
%o (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
%o (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
%Y Cf. A006093, A006530, A052126, A339464.
%Y Cf. A074781 (ratio=2^k), A339465 (ratio=2^q*3^r), A339463 (ratio=2^q*5^r).
%Y Cf. A122259.
%K nonn
%O 1,1
%A _Bernard Schott_, Dec 10 2020
%E More terms from _Amiram Eldar_, Dec 11 2020
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