

A257809


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


4



13, 67, 97, 139, 293, 661, 1163, 1657, 2039, 3203, 3469, 5171, 6361, 6661, 7393, 7901, 8969, 9103, 9137, 11971, 12301, 13487, 14083, 14699, 15473, 19141, 21247, 28099, 31039, 35423, 39047, 49223, 58427, 61493, 62171, 67699, 71971, 75869, 78857, 81533, 88007, 93199
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OFFSET

1,1


COMMENTS

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).
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...


LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000
Popular Computing (Calabasas, CA), The CSR Function, Vol. 4 (No. 34, Jan 1976), pages PC3410 to PC3411. Annotated and scanned copy.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC353 to PC354. Annotated and scanned copy.
Wikipedia, Feigenbaum constants.
1019 decimal digits of Feigenbaum's delta (due to David Broadhurst).


EXAMPLE

From M. F. Hasler, May 03 2018: (Start)
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...
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.
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)


PROG

(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


CROSSREFS

Cf. A006890, A257582, A257764, A257574.
Sequence in context: A058380 A129746 A067863 * A106975 A086689 A141956
Adjacent sequences: A257806 A257807 A257808 * A257810 A257811 A257812


KEYWORD

nonn


AUTHOR

Chai Wah Wu, May 10 2015


EXTENSIONS

Edited by M. F. Hasler, May 02 2018


STATUS

approved



