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A344603 Triangular numbers whose Hamming weight is also triangular.

%I A344603 #13 Feb 19 2023 16:59:19
%S A344603 0,1,21,28,190,231,276,378,435,630,741,903,946,1326,1540,1596,1830,
%T A344603 1953,2016,2278,2701,4278,4465,5460,5778,5886,6328,6441,6670,6903,
%U A344603 8646,11026,11781,11935,12246,12720,15225,15400,15931,16471,17391,17578,17955,18336,20100
%N A344603 Triangular numbers whose Hamming weight is also triangular.
%H A344603 Alois P. Heinz, <a href="/A344603/b344603.txt">Table of n, a(n) for n = 1..20000</a>
%H A344603 Audrey Baumheckel and Tamás Forgács, <a href="https://arxiv.org/abs/2105.10354">Guided by the primes -- an exploration of very triangular numbers</a>, arXiv:2105.10354 [math.HO], 2021.
%p A344603 q:= n-> issqr(8*add(i, i=Bits[Split](n))+1):
%p A344603 select(q, [j*(j+1)/2$j=0..200])[];  # _Alois P. Heinz_, May 24 2021
%t A344603 Select[Table[n*(n + 1)/2, {n, 0, 200}], IntegerQ @ Sqrt[8 * Plus @@ IntegerDigits[#, 2] + 1] &] (* _Amiram Eldar_, May 24 2021 *)
%t A344603 Select[Accumulate[Range[0,200]],OddQ[Sqrt[8 DigitCount[#,2,1]+1]]&] (* _Harvey P. Dale_, Feb 19 2023 *)
%o A344603 (PARI) isok(n) = ispolygonal(n, 3) && ispolygonal(hammingweight(n), 3);
%Y A344603 Cf. A000120.
%Y A344603 Intersection of A000217 and A344602.
%K A344603 nonn,base
%O A344603 1,3
%A A344603 _Michel Marcus_, May 24 2021