A069790 Triangular numbers with arithmetic mean of digits = 1 (sum of digits = number of digits).

1, 120, 210, 300, 112101, 100600020, 101111310, 110120220, 200130021, 200310120, 1000051003, 1010004040, 1130002030, 1411000003, 2002021003, 3200200003, 5000050000, 100110002070, 111111101310, 111202101003, 180000300000, 211104100200, 231201020001, 500001500001, 501001000500, 100021000424010

Offset: 1

Author

Amarnath Murthy, Apr 08 2002

Keywords

• base
• nonn

Comments

The sum of the digits of a triangular number is 0, 1, 3 or 6 (mod 9).

From Robert Israel, Aug 24 2018: (Start)

Suppose A007953(x) + A007953(2*x^2) - A055642(2*x^2) is even and

A007953(x) + A007953(2*x^2) >= 2*A055642(x) + A055642(2*x^2).

Then 10^k*x*(1+2*10^k*x) is in the sequence, where k = (A007953(x) + A007953(2*x^2) - A055642(2*x^2))/2.

In particular, x = 10^j-2 satisfies this criterion for all j>=1, with k = j. Thus the sequence is infinite. (End)

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..341 (all terms < 10^23)

Crossrefs

Cf. A007953, A055642.

Programs

Maple

T:= proc(n,k) option remember;
  if n*9 < k then return {} fi;
  if n = 1 then return {k} fi;
  `union`(seq(map(t -> 10*t+j, procname(n-1,k-j)),j=0..min(9,k)))
end proc:
T(1,0):= {}:
sort(convert(select(t -> issqr(8*t+1), `union`(seq(seq(T(9*i+j,9*i+j),j=[0,1,3,6]),i=0..1))),list)); # Robert Israel, Aug 24 2018

Mathematica

s=Select[Range[500000], Length[z=IntegerDigits[ #(#+1)/2]]==Plus@@z&]; s(s+1)/2
Select[Accumulate[Range[500000]],Mean[IntegerDigits[#]]==1&] (* Harvey P. Dale, May 05 2011 *)

Extensions

Edited by Dean Hickerson and Robert G. Wilson v, Apr 10 2002

More terms from Robert Israel, Aug 24 2018