%I #31 May 03 2019 08:40:58

%S 1,6,15,465,666,23436,93528,198765,664128,1493856,1786995,2767128,

%T 2953665,18292176,23891328,44655975,169878528,787667895,859984128,

%U 1934948736,3333238776,97844944896,237295393965,292957233975,379244291328,175847359339575,12999674534178816

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

%C 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.

%H Chai Wah Wu, <a href="/A307812/b307812.txt">Table of n, a(n) for n = 1..45</a> (all terms < 10^49. n = 1..31 from Sean A. Irvine, n = 32..35 from Giovanni Resta.)

%H C. Pomerance and Ch. Spicer, <a href="https://math.dartmouth.edu/~carlp/sheldon02132019.pdf">Proof of the Sheldon Conjecture</a>.

%e 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.

%t 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 *)

%Y Cf. A000217, A007954, A052382, A307792.

%K base,nonn

%O 1,2

%A _Ctibor O. Zizka_, Apr 30 2019

%E More terms from _Vaclav Kotesovec_, Apr 30 2019

%E a(26)-a(27) from _Chai Wah Wu_, May 01 2019