A349729 Numbers k >= 1 such that A018804(k) + A000203(k) is a triangular number.
2, 4, 5, 7, 10, 33, 34, 38, 49, 60, 92, 116, 132, 155, 159, 220, 268, 285, 315, 360, 437, 472, 579, 602, 664, 722, 835, 1254, 1269, 1320, 1336, 1348, 1436, 1786, 1797, 1890, 1996, 2016, 2024, 2050, 2115, 2163, 2344, 2427, 2455, 2595, 2710, 2961, 3497
Offset: 1
Keywords
Examples
k = 10 : A018804(10) = 27, A000203(10) = 18, 27 + 18 = 45 which is a triangular number thus 10 is a term.
Programs
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Mathematica
f1[p_, e_] := (p^(e + 1) - 1)/(p - 1); f2[p_, e_] := (e*(p - 1)/p + 1)*p^e; triQ[n_] := IntegerQ@Sqrt[8*n + 1]; q[n_] := triQ[Times @@ f1 @@@ (fct = FactorInteger[n]) + Times @@ f2 @@@ fct]; Select[Range[2, 3500], q] (* Amiram Eldar, Nov 27 2021 *)
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PARI
isok(k) = ispolygonal(sumdiv(k, d, k*eulerphi(d)/d) + sigma(k), 3); \\ Michel Marcus, Nov 27 2021