A374694 Primes that occur more than once in A373464.

8231, 10289, 10499, 15551, 20249, 40499, 49391, 51449, 59581, 96667, 117911, 123479, 152249, 159013, 161999, 165887, 239999, 255551, 257249, 260999, 288077, 292667, 314927, 319439, 453961, 514499, 519089, 524287, 524789, 530711, 565247, 580607, 657017, 774143

Offset

1,1

Comments

p is a term if and only if there exists more than one quadruple of primes (a,b,c,d) where d = p and (a+1,b+1,c+1,d+1) is a geometric progression.

Terms with 3 quadruples of primes are 59581, 161999, 255551, 292667, 530711, 580607, 657017, 1000187, 1427999, 1609631, 1718749, 2057999, ...

Terms with 4 quadruples of primes are 519089, 4991249, 5446237, ...

Terms with 5 quadruples of primes are 4393999, ...

Links

Chai Wah Wu, Table of n, a(n) for n = 1..192

Example

8231 is a term since (2, 41, 587, 8231) and (647, 1511, 3527, 8231) are quadruples of primes and (2+1, 41+1, 587+1, 8231+1) and (647+1, 1511+1, 3527+1, 8231+1) are geometric progressions.
10289 is a term since (239, 839, 2939, 10289) and (809, 1889, 4409, 10289) are quadruples of primes and (239+1, 839+1, 2939+1, 10289+1) and (809+1, 1889+1, 4409+1, 10289+1) are geometric progressions.

Prog

(Python)
from itertools import islice
from fractions import Fraction
from sympy import nextprime
def A374694_gen(): # generator of terms
    p, plist, pset = 1, [], set()
    while True:
        p = nextprime(p)
        flag = False
        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:
                        if flag:
                            yield p
                            break
                        flag = True
        plist = [p]+plist
        pset.add(p)
A374694_list = list(islice(A374694_gen(), 20))

Crossrefs

Cf. A373464.

Sequence in context: A031844 A210008 A168632 * A230043 A045056 A221482

Adjacent sequences: A374690 A374691 A374693 * A374695 A374696 A374697

Keyword

nonn

Author

Chai Wah Wu, Jul 16 2024

Status

approved