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A374694
Primes that occur more than once in A373464.
1
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
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
KEYWORD
nonn
AUTHOR
Chai Wah Wu, Jul 16 2024
STATUS
approved