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A373464
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Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.
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2
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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
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OFFSET
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1,1
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COMMENTS
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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.
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LINKS
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Doddy Kastanya, Fun Math #241, Number Theory group on LinkedIn.com, Jul 04 2024
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EXAMPLE
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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 = ...
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PROG
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(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)
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CROSSREFS
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Subsequence of A089199 (primes p such that p+1 is divisible by a cube).
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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