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