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A337156
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Numbers k such that the k-th triangular number has all its prime factors congruent to 1 mod 4.
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2
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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
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OFFSET
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1,2
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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lst={}; Do[p=1+8n; If[Union@Mod[First/@FactorInteger[p(p+1)/2], 4]=={1}, AppendTo[lst, p]], {n, 0, 10^3}]; lst
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PROG
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(PARI) isok(k) = my(f=factor(k*(k+1)/2)[, 1]~); #select(x->((x%4)==1), f) == #f; \\ Michel Marcus, Nov 22 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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