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

1,2

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.

Links

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

Example

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.

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

Prog

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

Crossrefs

Cf. A000217, A004613.

Sequence in context: A182323 A255184 A235941 * A069190 A124718 A126379

Adjacent sequences: A337153 A337154 A337155 * A337157 A337158 A337159

Keyword

nonn

Author

Frank M Jackson, Nov 21 2020

Status

approved