A267858 Positions of entries of A002972 that are congruent to 1 modulo 4.

1, 3, 4, 5, 6, 8, 10, 11, 12, 18, 19, 21, 23, 25, 26, 27, 28, 29, 30, 32, 33, 34, 36, 38, 41, 43, 45, 47, 49, 50, 52, 53, 55, 56, 57, 59, 60, 63, 65, 66, 68, 69, 72, 73, 74, 77, 78, 85, 87, 88, 89, 90, 91, 93, 94, 95, 96, 100, 104, 105, 106, 108, 110, 112, 115, 119, 120, 122, 127, 128, 131

Offset: 1

Author

Wolfdieter Lang, Feb 06 2016

Keywords

• nonn

Comments

This sequence is needed for the number of solutions modulo primes congruent to 1 modulo 4 of the elliptic curve y^2 = x^3 + x See A095978.

If a positive integer m is not in this sequence then A002972(m) == 3 (mod 4).

Examples

			n=1: A002972(1) = 1 == 1 (mod 4). But because m = 2 is not in this sequence A002972(2) = 3 == 3 (mod 4).
		

Crossrefs

Cf. A002145, A002972.

Programs

Mathematica

pmax = 2000; odd[p_] := Module[{k, m}, 2m+1 /. ToRules[Reduce[k>0 && m >= 0 && (2k)^2 + (2m+1)^2 == p, {k, m}, Integers]]]; Reap[For[n=1; p=5, p < pmax, p = NextPrime[p], If[Mod[p, 4]==1, If[Mod[odd[p], 4]==1, Sow[n]]; n++]]][[2, 1]] (* Jean-François Alcover, Feb 26 2016 *)

Formula

A002972(a(n)) == 1 (mod 4), n >= 1.