cp's OEIS Frontend

A374694 Primes that occur more than once in A373464.

%I A374694 #19 Jul 19 2024 14:31:52
%S A374694 8231,10289,10499,15551,20249,40499,49391,51449,59581,96667,117911,
%T A374694 123479,152249,159013,161999,165887,239999,255551,257249,260999,
%U A374694 288077,292667,314927,319439,453961,514499,519089,524287,524789,530711,565247,580607,657017,774143
%N A374694 Primes that occur more than once in A373464.
%C A374694 p is a term if and only if there exists more than one quadruple of primes (a,b,c,d) where d = p and (a+1,b+1,c+1,d+1) is a geometric progression.
%C A374694 Terms with 3 quadruples of primes are 59581, 161999, 255551, 292667, 530711, 580607, 657017, 1000187, 1427999, 1609631, 1718749, 2057999, ...
%C A374694 Terms with 4 quadruples of primes are 519089, 4991249, 5446237, ...
%C A374694 Terms with 5 quadruples of primes are 4393999, ...
%H A374694 Chai Wah Wu, <a href="/A374694/b374694.txt">Table of n, a(n) for n = 1..192</a>
%e A374694 8231 is a term since (2, 41, 587, 8231) and (647, 1511, 3527, 8231) are quadruples of primes and (2+1, 41+1, 587+1, 8231+1) and (647+1, 1511+1, 3527+1, 8231+1) are geometric progressions.
%e A374694 10289 is a term since (239, 839, 2939, 10289) and (809, 1889, 4409, 10289) are quadruples of primes and (239+1, 839+1, 2939+1, 10289+1) and (809+1, 1889+1, 4409+1, 10289+1) are geometric progressions.
%o A374694 (Python)
%o A374694 from itertools import islice
%o A374694 from fractions import Fraction
%o A374694 from sympy import nextprime
%o A374694 def A374694_gen(): # generator of terms
%o A374694     p, plist, pset = 1, [], set()
%o A374694     while True:
%o A374694         p = nextprime(p)
%o A374694         flag = False
%o A374694         for q in plist:
%o A374694             r = Fraction(q+1,p+1)
%o A374694             q2 = r*(q+1)-1
%o A374694             if q2 < 2:
%o A374694                 break
%o A374694             if q2.denominator == 1:
%o A374694                 q2 = int(q2)
%o A374694                 if q2 in pset:
%o A374694                     q3 = r*(q2+1)-1
%o A374694                     if q3 < 2:
%o A374694                         break
%o A374694                     if q3.denominator == 1 and int(q3) in pset:
%o A374694                         if flag:
%o A374694                             yield p
%o A374694                             break
%o A374694                         flag = True
%o A374694         plist = [p]+plist
%o A374694         pset.add(p)
%o A374694 A374694_list = list(islice(A374694_gen(),20))
%Y A374694 Cf. A373464.
%K A374694 nonn
%O A374694 1,1
%A A374694 _Chai Wah Wu_, Jul 16 2024