A329704 Numbers k such that the sum of divisors of k (A000203) and the sum of proper divisors of k (A001065) are both triangular numbers (A000217).

1, 2, 5, 36, 54, 441, 473, 6525, 52577, 124025, 683820, 1513754, 1920552, 6079931, 6762923, 14751657, 17052782, 17310942, 36543714, 49919939, 60260967, 251849052, 364535720, 372476909, 562047389, 670395564, 670440852, 783856979, 824626800, 1084201689, 1122603809

Offset: 1

Author

Amiram Eldar, Feb 28 2020

Keywords

• nonn

Comments

Are 1 and 36 the only terms that are also triangular numbers?

No other triangular terms up to A000217(10^8). - Michel Marcus, Mar 01 2020

Examples

			5 is a term since sigma(5) = 6 and sigma(5) - 5 = 1 are both triangular numbers.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..48

Crossrefs

Intersection of A045745 and A045746.

Cf. A000203, A000217, A001065.

Programs

Mathematica

triQ[n_] := IntegerQ @ Sqrt[8*n+1]; Select[Range[10^5], triQ[(s = DivisorSigma[1, #])] && triQ[s - #] &]

PARI

isok(k) = my(s=sigma(k)); ispolygonal(s, 3) && ispolygonal(s-k, 3); \\ Michel Marcus, Feb 29 2020