A350756 Integers whose number of divisors that are triangular numbers sets a new record.

1, 3, 6, 30, 90, 180, 210, 420, 630, 1260, 2520, 6930, 13860, 27720, 41580, 83160, 138600, 180180, 360360, 540540, 1081080, 1413720, 2162160, 3063060, 6126120, 12252240, 18378360, 36756720, 73513440, 91891800, 116396280, 183783600, 232792560, 349188840

Offset: 1

Author

Bernard Schott, Jan 13 2022

Keywords

• nonn

Comments

Terms that are triangular: 1, 3, 6, 210, 630, 2162160, ...

The number of triangular divisors of a(n) is A007862(a(n)): 1, 2, 3, 5, 6, 7, 8, 9, 10, 12, ...

Examples

			1260 has 36 divisors of which 12 are triangular numbers {1, 3, 6, 10, 15, 21, 28, 36, 45, 105, 210, 630}. No positive integer smaller than 1260 has as many as twelve triangular divisors; hence 1260 is a term.
		

Crossrefs

Cf. A000217, A007862, A130317.

Similar for A046952 (squares), A053624 (odd), A093036 (palindromes), A181808 (even), A340548 (repdigits), A340549 (repunits) divisors.

Programs

Mathematica

max=0;Do[If[(d=Length@Select[Divisors@k,IntegerQ[(Sqrt[8#+1]-1)/2]&])>max,Print@k;max=d],{k,10^10}] (* Giorgos Kalogeropoulos, Jan 13 2022 *)

PARI

lista(nn) = {my(r=0); for (n=1, nn, my(m = sumdiv(n, d, ispolygonal(d,3))); if (m>r, r=m; print1(n", ")));} \\ Michel Marcus, Jan 14 2022