A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.

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

1,2

Links

Table of n, a(n) for n=1..34.

C. K. Caldwell, Cunningham Chains

W. Roonguthai, Yves Gallot's Proth.exe and Cunningham Chains

Example

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.

Mathematica

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}]

Prog

(Python)
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

Crossrefs

Cf. A023272, A023302, A023330, A005602, A007700, A053176, A059452-A059456, A059500, A057331, A059688, A007053, A036378, A029837, A007053.

Sequence in context: A291294 A014208 A243358 * A330310 A121256 A022867

Adjacent sequences: A059687 A059688 A059689 * A059691 A059692 A059693

Keyword

nonn,more

Author

Labos Elemer, Feb 06 2001

Extensions

Edited and extended by Robert G. Wilson v, Nov 23 2002

Title and a(30)-a(31) corrected, and a(32) from Sean A. Irvine, Oct 02 2022

a(33)-a(34) from Michael S. Branicky, Oct 09 2022

Status

approved