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

1,2

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. - Robert Israel, Aug 24 2018

Links

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

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 *)

Crossrefs

Cf. A007953, A055642.

Sequence in context: A056994 A288461 A114823 * A064224 A069674 A003015

Adjacent sequences: A069787 A069788 A069789 * A069791 A069792 A069793

Keyword

base,nonn

Author

Amarnath Murthy, Apr 08 2002

Extensions

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

More terms from Robert Israel, Aug 24 2018

Status

approved