A059766
Initial (unsafe) primes of Cunningham chains of first type with length exactly 6.
Original entry on oeis.org
89, 63419, 127139, 405269, 810809, 1069199, 1178609, 1333889, 1598699, 1806089, 1958249, 2606069, 2848949, 3241289, 3339989, 3784199, 3962039, 4088879, 4444829, 4664249, 4894889, 4897709, 5132999, 5215499, 5238179, 6026309, 6059519, 6088529, 6490769, 6676259
Offset: 1
89 is a term because (89-1)/2 = 44 and 64*89+63 = 5759 = 13*443 are composites, while 89, 179, 359, 719, 1439, and 2879 are primes.
1122659 is not a term because it initiates a chain of length 7.
4658939 is not a term because (4658939-1)/2 = 2329469 is prime. - _Sean A. Irvine_, Oct 09 2022
Cf.
A023272,
A023302,
A023330,
A005384,
A005385,
A059452,
A059453,
A059454,
A059455,
A007700,
A059759,
A059760,
A059761,
A059762,
A059763,
A059764,
A038397,
A104349,
A091314,
A069362,
A016093,
A014937,
A057326.
A059690
Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.
Original entry on oeis.org
1, 2, 2, 2, 3, 5, 7, 13, 20, 31, 52, 83, 142, 242, 412, 742, 1308, 2294, 4040, 7327, 13253, 24255, 44306, 81700, 150401, 277335, 513705, 954847, 1780466, 3325109, 6224282, 11676337, 21947583, 41327438
Offset: 1
a(11)-a(10) = 21 means that between 1024 and 2048 exactly 21 primes introduce Cunningham chains: {1031, 1049, 1103, 1223, 1229, 1289, 1409, 1451, 1481, 1499, 1511, 1559, 1583, 1601, 1733, 1811, 1889, 1901, 1931, 1973, 2003}.
Their lengths are 2, 3 or 4. Thus the complete chains spread over more than one binary size-zone: {1409, 2819, 5639, 11279}. The primes 1439 and 2879 also form a chain but 1439 is not at the beginning of that chain, 89 is.
Cf.
A023272,
A023302,
A023330,
A005602,
A007700,
A053176,
A059452-
A059456,
A059500,
A057331,
A059688,
A007053,
A036378,
A029837,
A007053.
-
c = 0; k = 1; Do[ While[k <= 2^n, If[ PrimeQ[k] && !PrimeQ[(k - 1)/2] && PrimeQ[2k + 1], c++ ]; k++ ]; Print[c], {n, 1, 29}]
-
from itertools import count, islice
from sympy import isprime, primerange
def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
def agen():
s = 1
for n in count(2):
yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
print(list(islice(agen(), 20))) # Michael S. Branicky, Oct 09 2022
Title and a(30)-a(31) corrected, and a(32) from
Sean A. Irvine, Oct 02 2022
Showing 1-2 of 2 results.
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