A309812 Odd integers k such that k^2 is arithmetic mean of two other perfect squares.

5, 13, 15, 17, 25, 29, 35, 37, 39, 41, 45, 51, 53, 55, 61, 65, 73, 75, 85, 87, 89, 91, 95, 97, 101, 105, 109, 111, 113, 115, 117, 119, 123, 125, 135, 137, 143, 145, 149, 153, 155, 157, 159, 165, 169, 173, 175, 181, 183, 185, 187, 193, 195, 197, 203, 205, 215, 219

Offset

1,1

Comments

Odd numbers with at least one prime factor == 1 (mod 4). - Robert Israel, Oct 19 2025

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

5 is a term because 5^2 = 25 = (1^2 + 7^2)/2.

Maple

filter:= proc(n) ormap(t -> t mod 4 = 1, numtheory:-factorset(n)) end proc:
select(filter, [seq(i, i=1..1000, 2)]); # Robert Israel, Oct 19 2025

Mathematica

Select[Range[1, 300, 2], SquaresR[2, 2 #^2] > 4 &] (* Giovanni Resta, Aug 18 2019 *)
Select[Range[1, 300, 2], FactorInteger[#] != FactorInteger[#, GaussianIntegers -> True] &] (* Oliver Seipel, Oct 18 2025 *)

Prog

(PARI) isok(n) = {if (n %2, for (i=1, n, x = 2*n^2-i^2; if ((x!=i^2) && (x>0) && issquare(x), return (i)); ); ); } \\ Michel Marcus, Aug 18 2019

Crossrefs

Intersection of A005408 and A009003.

Sequence in context: A090759 A090758 A327922 * A324909 A322105 A230503

Adjacent sequences: A309809 A309810 A309811 * A309813 A309814 A309815

Keyword

nonn

Author

Mohsin A. Shaikh, Aug 18 2019

Extensions

More terms from Giovanni Resta, Aug 18 2019

Status

approved