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A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.

%I A373464 #34 Jul 18 2024 08:54:52
%S A373464 23,47,107,191,499,647,719,809,863,1249,1439,1999,2591,2879,3023,3779,
%T A373464 4079,5323,6911,7039,7127,7559,8231,8231,8747,9839,10289,10289,10499,
%U A373464 10499,10529,10691,11279,11519,12959,13229,13309,13999,15551,15551,15971,18143,19207
%N A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.
%C A373464 a(10) = 1249 is the first term not in A299171, a(15) = 3023 is the first term not in A293194, a(17) = 4079 is the first term not in A347977 and also the first term not in A374482, and a(21) = 7127 is the first term not in A184856.
%H A373464 Chai Wah Wu, <a href="/A373464/b373464.txt">Table of n, a(n) for n = 1..2372</a>
%H A373464 Doddy Kastanya, <a href="https://www.linkedin.com/feed/update/urn:li:activity:7214455399565938688">Fun Math #241</a>, Number Theory group on LinkedIn.com, Jul 04 2024
%e A373464 The terms of the sequence are column "p[4]" in the following table which lists the sequences of primes, and ratios of the geometric progression (p[k]+1):
%e A373464    n  | p[1], p[2], p[3], p[4]  |  r = (p[k+1]+1) / (p[k]+1)
%e A373464 ------+-------------------------+---------------------------
%e A373464    1  |    2,    5,   11,   23  |  2 = 6/3 = 12/6 = 24/12
%e A373464    2  |    5,   11,   23,   47  |  2 = 12/6 = 24/12 = 48/24
%e A373464    3  |   31,   47,   71,  107  |  3/2 = 48/32 = 72/48 = 108/72
%e A373464    4  |    2,   11,   47,  191  |  4 = 12/3 = 48/12 = 192/48
%e A373464    5  |   31,   79,  199,  499  |  5/2 = 80/32 = 200/80 = 500/200
%e A373464    6  |    2,   17,  107,  647  |  6 = 18/3 = 108/18 = 648/108
%e A373464    7  |   89,  179,  359,  719  |  2 = 180/90 = ...
%e A373464    8  |   29,   89,  269,  809  |  3 = 90/30 = ...
%e A373464    9  |  499,  599,  719,  863  |  6/5 = 600/500 = ...
%e A373464   10  |   79,  199,  499, 1249  |  5/2 = 200/80 = ...
%e A373464   11  |  179,  359,  719, 1439  |  2 = 360/180 = ...
%e A373464   12  |   53,  179,  599, 1999  |  10/3 = 180/54 = ...
%o A373464 (PARI) A373464_upto(N, show=0, D = 1, LIM=N\2) = { my(L=List()); forprime(p=1, LIM, my(denom = p+D); for(numer=denom+1, sqrtnint((N+D) * denom^2, 3), my(r=numer/denom); for(k=1,3, (type(denom * r^k)=="t_INT" && isprime(denom * r^k - D)) || next(2)); listput(L, denom * r^3 - D); show && printf(" | %4d, %4d, %4d, %4d | %s\n",denom-D, denom*r-D, denom*r^2-D, denom*r^3-D, numer/denom))); vecsort(L)}
%o A373464 (Python)
%o A373464 from itertools import islice
%o A373464 from fractions import Fraction
%o A373464 from sympy import nextprime
%o A373464 def A373464_gen(): # generator of terms
%o A373464     p, plist, pset = 1, [], set()
%o A373464     while True:
%o A373464         p = nextprime(p)
%o A373464         for q in plist:
%o A373464             r = Fraction(q+1,p+1)
%o A373464             q2 = r*(q+1)-1
%o A373464             if q2 < 2:
%o A373464                 break
%o A373464             if q2.denominator == 1:
%o A373464                 q2 = int(q2)
%o A373464                 if q2 in pset:
%o A373464                     q3 = r*(q2+1)-1
%o A373464                     if q3 < 2:
%o A373464                         break
%o A373464                     if q3.denominator == 1 and int(q3) in pset:
%o A373464                         yield p
%o A373464         plist = [p]+plist
%o A373464         pset.add(p)
%o A373464 A373464_list = list(islice(A373464_gen(),20)) # _Chai Wah Wu_, Jul 16 2024
%Y A373464 Subsequence of A089199 (primes p such that p+1 is divisible by a cube).
%K A373464 nonn
%O A373464 1,1
%A A373464 _M. F. Hasler_, Jul 12 2024
%E A373464 a(26)-a(43) from _Chai Wah Wu_, Jul 16 2024