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A338946 Lengths of Cunningham chains of the second kind that are sorted by first prime in the chain.

%I A338946 #11 Jun 30 2025 04:23:26
%S A338946 3,2,1,1,3,1,1,2,1,1,1,1,1,1,1,3,1,1,2,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,
%T A338946 1,1,1,2,2,1,1,2,1,1,1,1,1,1,1,2,1,1,1,2,1,1,3,2,1,1,1,1,2,1,2,1,1,1,
%U A338946 1,1,1,1,3,1,1,1,1,1,1,1,1,3,1,1,1,1,2,1,1,1,1,2,1,1,1,2,2,1,3,1,1,1,1,1,1
%N A338946 Lengths of Cunningham chains of the second kind that are sorted by first prime in the chain.
%C A338946 Row lengths of A338944.
%H A338946 Michael De Vlieger, <a href="/A338946/b338946.txt">Table of n, a(n) for n = 1..10000</a>
%H A338946 Chris K. Caldwell, <a href="https://primes.utm.edu/glossary/page.php?sort=CunninghamChain">Cunningham Chain</a> (PrimePages, Prime Glossary).
%H A338946 Wikipedia, <a href="https://en.wikipedia.org/wiki/Cunningham_chain">Cunningham chain</a>.
%e A338946 We start with p = 2. Since 2(2) - 1 = 3 is prime, and further 2(3) - 1 = 5 is prime, but 2(5) - 1 is composite, we have chain length 3, so a(1) = 3.
%e A338946 p = 7 is the smallest prime that hasn't appeared in a chain thus far; since 2(7) - 1 = 13 is prime but 2(13) - 1 = 25 is composite, we have a chain of length 2, so a(2) = 2.
%e A338946 p = 11 is the smallest prime that hasn't appeared in a chain; 2(11) - 1 = 21 is composite, so we have a singleton chain, thus a(3) = 1, etc.
%t A338946 Block[{a = {2}, b = {}, j = 0, k, p}, Do[k = Length@ b + 1; If[PrimeQ@ a[[-1]], AppendTo[a, 2 a[[-1]] - 1]; j++, While[! FreeQ[a, Set[p, Prime[k]]], k++]; AppendTo[b, j]; Set[j, 0]; Set[a, Append[a[[1 ;; -2]], p]]], {10^3}]; b]
%Y A338946 Cf. A075712, A338944, A338945.
%K A338946 nonn
%O A338946 1,1
%A A338946 _Michael De Vlieger_, Nov 17 2020