A337156 Numbers k such that the k-th triangular number has all its prime factors congruent to 1 mod 4.

1, 25, 73, 145, 169, 193, 289, 313, 337, 409, 457, 481, 577, 625, 673, 697, 745, 793, 841, 865, 985, 1009, 1129, 1153, 1201, 1249, 1321, 1345, 1369, 1417, 1465, 1489, 1513, 1537, 1585, 1657, 1681, 1753, 1801, 1873, 1993, 2017, 2041, 2137, 2257, 2305, 2329, 2377, 2425, 2473

Offset: 1

Author

Frank M Jackson, Nov 21 2020

Keywords

• nonn

Comments

The k-th triangular number t_k is given as t_k = k(k+1)/2. The t_k associated with this sequence form the intersection of A004613 and A000217.

Apart from 1, numbers whose prime factors are all congruent to 1 mod 4 are also known as primitive hypotenuse numbers because they are candidates for the hypotenuse of primitive right triangles.

For t_k to be a primitive hypotenuse number all its divisors must be congruent to 1 mod 4. Therefore k has to be odd and congruent to 1 mod 8.

Examples

			a(2) = 25 because the 25th triangular number is 325, the prime factorization of 325 is 5^2*13, and 5,13 are both congruent to 1 mod 4. It is the second such occurrence.
		

Links

• Frank M Jackson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000217, A004613.

Programs

Mathematica

lst={}; Do[p=1+8n;If[Union@Mod[First/@FactorInteger[p(p+1)/2], 4]=={1}, AppendTo[lst, p]], {n, 0, 10^3}]; lst
Select[Range[2500],Union[Mod[FactorInteger[(#(#+1))/2][[;;,1]],4]]=={1}&] (* Harvey P. Dale, Aug 07 2025 *)

PARI

isok(k) = my(f=factor(k*(k+1)/2)[,1]~); #select(x->((x%4)==1), f) == #f; \\ Michel Marcus, Nov 22 2020