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A100208
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Minimal permutation of the natural numbers such that the sum of squares of two consecutive terms is a prime.
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14
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1, 2, 3, 8, 5, 4, 9, 10, 7, 12, 13, 20, 11, 6, 19, 14, 15, 22, 17, 18, 23, 30, 29, 16, 25, 24, 35, 26, 21, 34, 39, 40, 33, 28, 37, 32, 27, 50, 31, 44, 41, 46, 49, 36, 65, 38, 45, 52, 57, 68, 43, 42, 55, 58, 47, 48, 53, 62, 73, 60, 61, 54, 59, 64, 71, 66, 79, 56, 51, 76, 85, 72
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OFFSET
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1,2
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COMMENTS
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a(1) = 1 and for n>1: a(n) = smallest m not occurring earlier such that m^2 + a(n-1)^2 is a prime; the primes are in A100209.
Note the same parity of a(n) and n for all terms. [Zak Seidov, Apr 27 2011]
Subsequence s(1..m) is a permutation of the natural numbers 1..m only for m=1,2,3. [Zak Seidov, Apr 28 2011]
All filtering primes (A100209) are distinct because primes of the form 4k+1 have a unique representation as the sum of two squares. [Zak Seidov, Apr 28 2011]
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LINKS
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FORMULA
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MATHEMATICA
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nn = 100; unused = Range[2, nn]; t = {1}; While[k = 0; While[k++; k <= Length[unused] && ! PrimeQ[t[[-1]]^2 + unused[[k]]^2]]; k <= Length[unused], AppendTo[t, unused[[k]]]; unused = Delete[unused, k]]; t (* T. D. Noe, Apr 27 2011 *)
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PROG
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(Haskell)
import Data.Set (singleton, notMember, insert)
a100208 n = a100208_list !! (n-1)
a100208_list = 1 : (f 1 [1..] $ singleton 1) where
f x (w:ws) s
| w `notMember` s &&
a010051 (x*x + w*w) == 1 = w : (f w [1..] $ insert w s)
| otherwise = f x ws s where
(Python)
from sympy import isprime
for n in range(1, 100):
while not isprime(b) or a in A100208:
b += 2*a+1
a += 1
(PARI) v=[1]; n=1; while(n<100, if(isprime(v[#v]^2+n^2)&&!vecsearch(vecsort(v), n), v=concat(v, n); n=0); n++); v \\ Derek Orr, Jun 01 2015
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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