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

1,1

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: A373461 A373462 A373463 * 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