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 PC34-10 to PC34-11. Annotated and scanned copy.
Herman P. Robinson, The CSR Function, Popular Computing (Calabasas, CA), Vol. 4 (No. 35, Feb 1976), pages PC35-3 to PC35-4. Annotated and scanned copy.
Wikipedia, Feigenbaum constants.
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
KEYWORD
nonn
AUTHOR
Chai Wah Wu, May 10 2015
EXTENSIONS
Edited by M. F. Hasler, May 02 2018
STATUS
approved