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

%I A059690 #16 Apr 03 2023 10:36:09
%S A059690 1,2,2,2,3,5,7,13,20,31,52,83,142,242,412,742,1308,2294,4040,7327,
%T A059690 13253,24255,44306,81700,150401,277335,513705,954847,1780466,3325109,
%U A059690 6224282,11676337,21947583,41327438
%N A059690 Number of distinct Cunningham chains of first kind whose initial prime (cf. A059453) <= 2^n.
%H A059690 C. K. Caldwell, <a href="https://t5k.org/glossary/page.php/CunninghamChain">Cunningham Chains</a>
%H A059690 W. Roonguthai, <a href="http://ksc9.th.com/warut/cunningham.html">Yves Gallot's Proth.exe and Cunningham Chains</a>
%e A059690 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}.
%e A059690 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.
%t A059690 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}]
%o A059690 (Python)
%o A059690 from itertools import count, islice
%o A059690 from sympy import isprime, primerange
%o A059690 def c(p): return not isprime((p-1)//2) and isprime(2*p+1)
%o A059690 def agen():
%o A059690     s = 1
%o A059690     for n in count(2):
%o A059690         yield s; s += sum(1 for p in primerange(2**(n-1)+1, 2**n) if c(p))
%o A059690 print(list(islice(agen(), 20))) # _Michael S. Branicky_, Oct 09 2022
%Y A059690 Cf. A023272, A023302, A023330, A005602, A007700, A053176, A059452-A059456, A059500, A057331, A059688, A007053, A036378, A029837, A007053.
%K A059690 nonn,more
%O A059690 1,2
%A A059690 _Labos Elemer_, Feb 06 2001
%E A059690 Edited and extended by _Robert G. Wilson v_, Nov 23 2002
%E A059690 Title and a(30)-a(31) corrected, and a(32) from _Sean A. Irvine_, Oct 02 2022
%E A059690 a(33)-a(34) from _Michael S. Branicky_, Oct 09 2022