A373464 Largest of a quadruple of primes p[1..4] such that (p[k]+1, k=1..4) is in geometric progression.
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
Offset: 1
Keywords
Examples
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 = ...
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
Crossrefs
Subsequence of A089199 (primes p such that p+1 is divisible by a cube).
Programs
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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)}
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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
Extensions
a(26)-a(43) from Chai Wah Wu, Jul 16 2024
Comments