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A257809 Lexicographically largest strictly increasing sequence of primes for which the continued square root map produces Feigenbaum's constant delta = 4.6692016... (A006890).

%I A257809 #19 May 05 2018 15:21:35
%S A257809 13,67,97,139,293,661,1163,1657,2039,3203,3469,5171,6361,6661,7393,
%T A257809 7901,8969,9103,9137,11971,12301,13487,14083,14699,15473,19141,21247,
%U A257809 28099,31039,35423,39047,49223,58427,61493,62171,67699,71971,75869,78857,81533,88007,93199
%N A257809 Lexicographically largest strictly increasing sequence of primes for which the continued square root map produces Feigenbaum's constant delta = 4.6692016... (A006890).
%C A257809 The continued square root map takes a finite or infinite sequence (x, y, z, ...) to the number CSR(x, y, z,...) = sqrt(x + sqrt(y + sqrt(z + ...))). It is well defined if the logarithm of the terms is O(2^n).
%C A257809 The terms are defined to be the largest possible choice such that the sequence can remain strictly increasing without the CSR growing beyond delta = 4.66920...
%H A257809 Chai Wah Wu, <a href="/A257809/b257809.txt">Table of n, a(n) for n = 1..1000</a>
%H A257809 Popular Computing (Calabasas, CA), <a href="/A257352/a257352.pdf">The CSR Function</a>, Vol. 4 (No. 34, Jan 1976), pages PC34-10 to PC34-11. Annotated and scanned copy.
%H A257809 Herman P. Robinson, <a href="/A257574/a257574.pdf">The CSR Function</a>, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
%H A257809 Wikipedia, <a href="http://en.wikipedia.org/wiki/Feigenbaum_constants">Feigenbaum constants</a>.
%H A257809 <a href="http://sprott.physics.wisc.edu/phys505/feigen.htm">1019 decimal digits of Feigenbaum's delta (due to David Broadhurst)</a>.
%e A257809 From _M. F. Hasler_, May 03 2018: (Start)
%e A257809 We look for a strictly increasing sequence of primes (p,q,r,...) such that CSR(p,q,r,...) = sqrt(p + sqrt(q + sqrt(r + ...))) = delta = 4.66920...
%e A257809 The first term must be less than delta^2 ~ 21.8, but p = 19 and also p = 17 are excluded, since CSR(17,19,23,...) > 4.67. It appears that p = 13 does not lead to a contradiction, so this is the largest possible choice for p, whence a(1) = 13.
%e A257809 The second term could be chosen to be q = 17, provided that subsequent terms are large enough to ensure CSR(p, q, r,...) = delta, which is always possible. But one can verify that any q between 19 and 67 is also possible without contradiction. If we try q = 71, then we find that CSR(13, 71, 73, ...) > 4.68. So a(2) = 67, etc. (End)
%o A257809 (PARI) (CSR(v,s)=forstep(i=#v,1,-1,s=sqrt(v[i]+s));s); a=[13]; for(n=1,50, print1(a[#a]","); for(i=primepi(a[#a])+1,oo, CSR(concat(a,vector(9,j,prime(i+j))))>=delta&& (a=concat(a,prime(i)))&& break)) \\  For delta, see A006890. - _M. F. Hasler_, May 03 2018
%Y A257809 Cf. A006890, A257582, A257764, A257574.
%K A257809 nonn
%O A257809 1,1
%A A257809 _Chai Wah Wu_, May 10 2015
%E A257809 Edited by _M. F. Hasler_, May 02 2018