A360061 Lexicographically earliest increasing sequence such that a(1) = 2 and for n >= 2, a(1)^2 + a(2)^2 + ... + a(n)^2 is a prime.

2, 3, 4, 12, 48, 54, 66, 138, 144, 162, 168, 180, 198, 234, 252, 264, 330, 360, 366, 372, 402, 420, 444, 462, 480, 534, 546, 552, 564, 576, 600, 630, 642, 678, 702, 744, 756, 846, 852, 858, 882, 966, 1008, 1206, 1242, 1254, 1266, 1272, 1296, 1302, 1338, 1650

Offset: 1

Author

Win Wang, Jan 23 2023

Keywords

• easy
• nonn

Examples

			For n >= 2, partial sums of squares are (showing primality): 2^2 + 3^2 = 13; 13 + 4^2 = 29; 29 + 12^2 = 173; 173 + 48^2 = 2477; ...
		

Crossrefs

Cf. A051935, A137326.

Programs

Haskell

import Math.NumberTheory.Primes.Testing (isPrime)
a360061_list = 2 : 3 : recurse 4 13 where
  recurse n p
    | isPrime(n^2 + p) = n : recurse (n+1) (n^2 + p)
    | otherwise        = recurse (n+1) p
-- Peter Kagey, Jan 25 2023

Maple

s:= proc(n) option remember; `if`(n<1, 0, a(n)^2+s(n-1)) end:
a:= proc(n) option remember; local k, m;
      k:= s(n-1); for m from 1+a(n-1)
      while not isprime(k+m^2) do od; m
    end: a(1):=2:
seq(a(n), n=1..52);  # Alois P. Heinz, Jan 26 2023

Mathematica

s[n_] := s[n] = If[n < 1, 0, a[n]^2 + s[n-1]];
a[n_] := a[n] = Module[{k, m},
   k = s[n-1]; For[m = 1 + a[n-1],
   !PrimeQ[k + m^2], m++]; m];
a[1] = 2;
Table[a[n], {n, 1, 52}] (* Jean-François Alcover, Feb 03 2025, after Alois P. Heinz *)