A354722 Composite numbers whose divisors have distinct binary weights (A000120).

25, 39, 55, 57, 69, 87, 95, 111, 115, 119, 121, 123, 125, 141, 145, 159, 169, 177, 183, 185, 187, 201, 203, 205, 213, 215, 219, 221, 235, 237, 249, 253, 265, 289, 291, 299, 301, 303, 305, 319, 321, 323, 329, 335, 339, 355, 361, 365, 371, 377, 391, 393, 411, 413

Offset

1,1

Comments

Without the restriction of composite numbers, 1 and all the odd primes would have been terms of this sequence.

Since 1 and 2 have the same binary weight, all the terms are odd.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Example

25 is a term since its divisors, 1, 5 and 25, have binary weights 1, 2 and 3, respectively.
55 is a term since its divisors, 1, 5, 11 and 55, have binary weights 1, 2, 3 and 5, respectively.

Mathematica

bw[n_] := DigitCount[n, 2, 1]; q[n_] := CompositeQ[n] && UnsameQ @@ (bw /@ Divisors[n]); Select[Range[1, 400, 2], q]

Prog

(Python)
from sympy import divisors
def binwt(n): return bin(n).count("1")
def ok(n):
    binwts, divs = set(), 0
    for d in divisors(n, generator=True):
        b = binwt(d)
        if b in binwts: return False
        binwts.add(b)
        divs += 1
    return divs > 2
print([k for k in range(415) if ok(k)]) # Michael S. Branicky, Jun 04 2022
(PARI) isok(c) = {if ((c>1) && !isprime(c), my(d=divisors(c)); #Set(apply(hammingweight, d)) == #d; ); } \\ Michel Marcus, Jun 04 2022

Crossrefs

Cf. A000120, A027750.

Subsequences: A255401 and A354724.

Sequence in context: A080386 A070758 A104667 * A354723 A066844 A255608

Adjacent sequences: A354719 A354720 A354721 * A354723 A354724 A354725

Keyword

nonn,base

Author

Amiram Eldar, Jun 04 2022

Status

approved