A100208 Minimal permutation of the natural numbers such that the sum of squares of two consecutive terms is a prime.

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

Offset: 1

Author

Reinhard Zumkeller, Nov 08 2004

Keywords

• nonn
• easy

Comments

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]

Links

• Zak Seidov, Table of n, a(n) for n = 1..20000
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A100209, A100211, A171964, A181723, A181730 [Zak Seidov, Apr 27 2011].

Cf. A080478, A010051.

Programs

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
-- Reinhard Zumkeller, Apr 28 2011

Mathematica

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 *)

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

Python

from sympy import isprime
A100208 = [1]
for n in range(1,100):
    a, b = 1, 1 + A100208[-1]**2
    while not isprime(b) or a in A100208:
        b += 2*a+1
        a += 1
    A100208.append(a) # Chai Wah Wu, Sep 01 2014

Formula

a(A100211(n)) = A100211(a(n)) = n.

a(n) = sqrt(A073658(n)).

a(n)^2 + a(n+1)^2 = A100209(n).