A344602 -id:A344602 - cp's OEIS frontend

A344603 Triangular numbers whose Hamming weight is also triangular.

0, 1, 21, 28, 190, 231, 276, 378, 435, 630, 741, 903, 946, 1326, 1540, 1596, 1830, 1953, 2016, 2278, 2701, 4278, 4465, 5460, 5778, 5886, 6328, 6441, 6670, 6903, 8646, 11026, 11781, 11935, 12246, 12720, 15225, 15400, 15931, 16471, 17391, 17578, 17955, 18336, 20100

Offset: 1

Michel Marcus, May 24 2021

Alois P. Heinz, Table of n, a(n) for n = 1..20000
Audrey Baumheckel and Tamás Forgács, Guided by the primes -- an exploration of very triangular numbers, arXiv:2105.10354 [math.HO], 2021.

Cf. A000120.

Intersection of A000217 and A344602.

q:= n-> issqr(8*add(i, i=Bits[Split](n))+1):
select(q, [j*(j+1)/2$j=0..200])[];  # Alois P. Heinz, May 24 2021

Select[Table[n*(n + 1)/2, {n, 0, 200}], IntegerQ @ Sqrt[8 * Plus @@ IntegerDigits[#, 2] + 1] &] (* Amiram Eldar, May 24 2021 *)
Select[Accumulate[Range[0,200]],OddQ[Sqrt[8 DigitCount[#,2,1]+1]]&] (* Harvey P. Dale, Feb 19 2023 *)

isok(n) = ispolygonal(n, 3) && ispolygonal(hammingweight(n), 3);

A381720 Integers whose Hamming weight is a cube.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 32, 64, 128, 255, 256, 383, 447, 479, 495, 503, 507, 509, 510, 512, 639, 703, 735, 751, 759, 763, 765, 766, 831, 863, 879, 887, 891, 893, 894, 927, 943, 951, 955, 957, 958, 975, 983, 987, 989, 990, 999, 1003, 1005, 1006, 1011, 1013, 1014

Offset: 1

Ctibor O. Zizka, Mar 05 2025

The plot of terms of the form k - 2^floor(log_2(k)) shows quasi-periodic structures on the intervals [2^i, 2^(i+1)]. In general, all sequences of the form "Integers whose Hamming weight is f(x)", where f(x) is an integer valued function, are quasi-periodic on intervals [2^i, 2^(i+1)].
The powers of 2 (A000079) are terms.
A023690 is a subsequence.

			For k = 255: A000120(255) = 8 = 2^3 is a cube, thus 255 is a term.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000079, A000120, A000578, A023690.

Cf. A084561, A344602.

filter:= proc(n) type(surd(convert(convert(n,base,2),`+`),3),integer) end proc:
select(filter, [$0..2000]); # Robert Israel, May 18 2025

Select[Range[0, 1200], IntegerQ[Surd[DigitCount[#, 2, 1], 3]] &] (* Amiram Eldar, Mar 05 2025 *)

isok(k) = ispower(hammingweight(k), 3); \\ Michel Marcus, Mar 05 2025

from itertools import count, islice, combinations
from sympy import integer_nthroot
def A381720_gen(): # generator of terms
    a = []
    yield 0
    for l in count(1):
        b = 1<A381720_list = list(islice(A381720_gen(),53)) # Chai Wah Wu, Mar 06 2025

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A344603 Triangular numbers whose Hamming weight is also triangular.

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