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%I #13 Nov 27 2021 12:01:13
%S 2,4,5,7,10,33,34,38,49,60,92,116,132,155,159,220,268,285,315,360,437,
%T 472,579,602,664,722,835,1254,1269,1320,1336,1348,1436,1786,1797,1890,
%U 1996,2016,2024,2050,2115,2163,2344,2427,2455,2595,2710,2961,3497
%N Numbers k >= 1 such that A018804(k) + A000203(k) is a triangular number.
%e k = 10 : A018804(10) = 27, A000203(10) = 18, 27 + 18 = 45 which is a triangular number thus 10 is a term.
%t 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 *)
%o (PARI) isok(k) = ispolygonal(sumdiv(k, d, k*eulerphi(d)/d) + sigma(k), 3); \\ _Michel Marcus_, Nov 27 2021
%Y Cf. A000203, A000217, A018804.
%K nonn
%O 1,1
%A _Ctibor O. Zizka_, Nov 27 2021