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Triangular numbers that are repdigits with length > 2 in some base.
1

%I #10 Jun 10 2016 02:57:03

%S 15,21,91,171,666,703,820,1830,1953,3003,3081,4095,7140,7381,10440,

%T 12720,14706,16471,16653,18915,23871,24976,30628,47586,47895,48828,

%U 66430,71631,79401,95703,101475,104653,119805,128778,148240,148785,173166,191271,221445

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

%C Intersection of A000217 and A167782.

%C 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 Robert Israel, <a href="/A274084/b274084.txt">Table of n, a(n) for n = 1..694</a>

%e 15 = 5*6/2 = 1111_2.

%e 21 = 6*7/2 = 111_4.

%e 91 = 13*14/2 = 111_9.

%e 171 = 18*19/2 = 333_7.

%p N:= 10^9: # to get all entries <= N

%p S:= {}:

%p for b from 2 to floor(sqrt(N)) do

%p for k from 3 do

%p r:= (b^k-1)/(b-1);

%p if r > N then break fi;

%p for a from 1 to min(b-1, N/r) do

%p if issqr(1+8*r*a) then

%p S:= S union {r*a}

%p fi

%p od

%p od

%p od:

%p sort(convert(S,list));

%Y Cf. A000217, A045914, A167782.

%K nonn,base

%O 1,1

%A _Robert Israel_, Jun 09 2016