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A274084 Triangular numbers that are repdigits with length > 2 in some base.

%I A274084 #10 Jun 10 2016 02:57:03
%S A274084 15,21,91,171,666,703,820,1830,1953,3003,3081,4095,7140,7381,10440,
%T A274084 12720,14706,16471,16653,18915,23871,24976,30628,47586,47895,48828,
%U A274084 66430,71631,79401,95703,101475,104653,119805,128778,148240,148785,173166,191271,221445
%N A274084 Triangular numbers that are repdigits with length > 2 in some base.
%C A274084 Intersection of A000217 and A167782.
%C A274084 Sequence is infinite, e.g. for any k>=2 and j>=1 it contains n*(n+1)/2 where n = ((8j+1)^k-1)/2: this has 2k digits of j in base 8j+1.
%H A274084 Robert Israel, <a href="/A274084/b274084.txt">Table of n, a(n) for n = 1..694</a>
%e A274084 15 = 5*6/2 = 1111_2.
%e A274084 21 = 6*7/2 = 111_4.
%e A274084 91 = 13*14/2 = 111_9.
%e A274084 171 = 18*19/2 = 333_7.
%p A274084 N:= 10^9: # to get all entries <= N
%p A274084 S:= {}:
%p A274084 for b from 2 to floor(sqrt(N)) do
%p A274084   for k from 3 do
%p A274084      r:= (b^k-1)/(b-1);
%p A274084      if r > N then break fi;
%p A274084      for a from 1 to min(b-1, N/r) do
%p A274084         if issqr(1+8*r*a) then
%p A274084           S:= S union {r*a}
%p A274084         fi
%p A274084      od
%p A274084   od
%p A274084 od:
%p A274084 sort(convert(S,list));
%Y A274084 Cf. A000217, A045914, A167782.
%K A274084 nonn,base
%O A274084 1,1
%A A274084 _Robert Israel_, Jun 09 2016