A307812 Triangular numbers t(n) with a zeroless decimal representation such that (product of decimal digits of t(n)) / n is an integer.

1, 6, 15, 465, 666, 23436, 93528, 198765, 664128, 1493856, 1786995, 2767128, 2953665, 18292176, 23891328, 44655975, 169878528, 787667895, 859984128, 1934948736, 3333238776, 97844944896, 237295393965, 292957233975, 379244291328, 175847359339575, 12999674534178816

Offset

1,2

Comments

Are all terms of the sequence bigger than 1 divisible by 3? I conjecture that 1 and 15 are the only terms for which (product of decimal digits of t(n)) = n.

Links

Chai Wah Wu, Table of n, a(n) for n = 1..45 (all terms < 10^49. n = 1..31 from Sean A. Irvine, n = 32..35 from Giovanni Resta.)

C. Pomerance and Ch. Spicer, Proof of the Sheldon Conjecture.

Example

For n = 30, t(30) = 465, product of decimal digits of t(30) = 4*6*5 = 120, product of decimal digits of t(n) / n = 120 / 30 = 4 so t(30) = 465 is in the sequence.

Mathematica

idx = Select[Range[100000], Product[j, {j, IntegerDigits[#*(# + 1)/2]}] != 0 && Divisible[Product[j, {j, IntegerDigits[#*(# + 1)/2]}], #] &]; idx*(idx + 1)/2 (* Vaclav Kotesovec, Apr 30 2019 *)

Crossrefs

Cf. A000217, A007954, A052382, A307792.

Sequence in context: A024081 A145558 A145612 * A354682 A280964 A191463

Adjacent sequences: A307809 A307810 A307811 * A307813 A307814 A307815

Keyword

base,nonn

Author

Ctibor O. Zizka, Apr 30 2019

Extensions

More terms from Vaclav Kotesovec, Apr 30 2019

a(26)-a(27) from Chai Wah Wu, May 01 2019

Status

approved