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A257809
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Lexicographically largest strictly increasing sequence of primes for which the continued square root map produces Feigenbaum's constant delta = 4.6692016... (A006890).
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4
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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
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1,1
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COMMENTS
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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...
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LINKS
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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.
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EXAMPLE
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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)
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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