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

1,2

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, ...

Links

Table of n, a(n) for n=1..34.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A000217, A007862, A130317.

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

Sequence in context: A211168 A355989 A215294 * A090932 A361864 A280981

Adjacent sequences: A350753 A350754 A350755 * A350757 A350758 A350759

Keyword

nonn

Author

Bernard Schott, Jan 13 2022

Status

approved