A344602 Integers whose Hamming weight is triangular.

0, 1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, 32, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 63, 64, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 95, 97, 98, 100, 104, 111, 112, 119, 123, 125, 126, 128, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 159, 161

Offset

1,3

Links

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Maple

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

Mathematica

Select[Range[0, 200], IntegerQ @ Sqrt[8 * Plus @@ IntegerDigits[#, 2] + 1] &] (* Amiram Eldar, May 24 2021 *)

Prog

(PARI) isok(n) = ispolygonal(hammingweight(n), 3);
(Python)
from math import isqrt, comb
from sympy.ntheory.primetest import is_square
from oeis_sequences.OEISsequences import bisection
def A344602(n):
    def f(x):
        s = bin(x)[-1:1:-1]
        m = x.bit_count()
        l = x.bit_length()
        l2 = (k:=isqrt(t:=l+1<<1))+int((t<<2)>(k<<2)*(k+1)+1)
        a = is_square((m<<3)|1)
        c = n+x-a
        for i in range(l):
            j = int(s[i])
            if j:
                m -= 1
                r = (k:=isqrt(t:=m+1<<1))+int((t<<2)>(k<<2)*(k+1)+1)-1
                for p in range(r+(not is_square((m<<3)|1)), l2):
                    c -= comb(i, comb(p+1, 2)-m)
        return c
    return bisection(f, n-1, n-1) # Chai Wah Wu, Jun 09 2026

Crossrefs

Cf. A000120, A000217, A344603.

Sequence in context: A161989 A229829 A000069 * A333975 A140137 A080308

Adjacent sequences: A344599 A344600 A344601 * A344603 A344604 A344605

Keyword

nonn,base

Author

Michel Marcus, May 24 2021

Status

approved