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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. 2
 23, 47, 107, 191, 499, 647, 719, 809, 863, 1249, 1439, 1999, 2591, 2879, 3023, 3779, 4079, 5323, 6911, 7039, 7127, 7559, 8231, 8231, 8747, 9839, 10289, 10289, 10499, 10499, 10529, 10691, 11279, 11519, 12959, 13229, 13309, 13999, 15551, 15551, 15971, 18143, 19207 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS 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. LINKS Chai Wah Wu, Table of n, a(n) for n = 1..2372 Doddy Kastanya, Fun Math #241, Number Theory group on LinkedIn.com, Jul 04 2024 EXAMPLE 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): n | p[1], p[2], p[3], p[4] | r = (p[k+1]+1) / (p[k]+1) ------+-------------------------+--------------------------- 1 | 2, 5, 11, 23 | 2 = 6/3 = 12/6 = 24/12 2 | 5, 11, 23, 47 | 2 = 12/6 = 24/12 = 48/24 3 | 31, 47, 71, 107 | 3/2 = 48/32 = 72/48 = 108/72 4 | 2, 11, 47, 191 | 4 = 12/3 = 48/12 = 192/48 5 | 31, 79, 199, 499 | 5/2 = 80/32 = 200/80 = 500/200 6 | 2, 17, 107, 647 | 6 = 18/3 = 108/18 = 648/108 7 | 89, 179, 359, 719 | 2 = 180/90 = ... 8 | 29, 89, 269, 809 | 3 = 90/30 = ... 9 | 499, 599, 719, 863 | 6/5 = 600/500 = ... 10 | 79, 199, 499, 1249 | 5/2 = 200/80 = ... 11 | 179, 359, 719, 1439 | 2 = 360/180 = ... 12 | 53, 179, 599, 1999 | 10/3 = 180/54 = ... PROG (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)} (Python) from itertools import islice from fractions import Fraction from sympy import nextprime def A373464_gen(): # generator of terms p, plist, pset = 1, [], set() while True: p = nextprime(p) for q in plist: r = Fraction(q+1, p+1) q2 = r*(q+1)-1 if q2 < 2: break if q2.denominator == 1: q2 = int(q2) if q2 in pset: q3 = r*(q2+1)-1 if q3 < 2: break if q3.denominator == 1 and int(q3) in pset: yield p plist = [p]+plist pset.add(p) A373464_list = list(islice(A373464_gen(), 20)) # Chai Wah Wu, Jul 16 2024 CROSSREFS Subsequence of A089199 (primes p such that p+1 is divisible by a cube). Sequence in context: A039374 A043197 A043977 * A042048 A347008 A239563 Adjacent sequences: A373460 A373461 A373462 * A373465 A373466 A373467 KEYWORD nonn AUTHOR M. F. Hasler, Jul 12 2024 EXTENSIONS a(26)-a(43) from Chai Wah Wu, Jul 16 2024 STATUS approved

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Last modified August 3 20:33 EDT 2024. Contains 374905 sequences. (Running on oeis4.)