A358970 Nonnegative numbers m such that if 2^k appears in the binary expansion of m, then k+1 divides m.

0, 1, 2, 6, 8, 12, 36, 60, 128, 136, 168, 261, 288, 520, 530, 540, 630, 640, 1056, 2052, 2088, 2100, 2184, 2208, 2304, 2340, 2520, 2580, 4134, 8232, 8400, 8820, 9240, 10248, 10920, 16440, 16560, 16920, 16950, 17010, 17040, 17190, 17280, 18480, 18600, 18720

Offset

1,3

Comments

In other words, numbers whose binary expansion encodes a subset of their divisors.

Also numbers m divisible by A271410(m).

This sequence is infinite as it contains A058891.

Links

Table of n, a(n) for n=1..46.

Index entries for sequences related to binary expansion of n

Example

60 = 2^5 + 2^4 + 2^3 + 2^2 and 60 is divisible by 5+1, 4+1, 3+1 and 2+1, so 60 belongs to the sequence.
42 = 2^5 + 2^3 + 2^1 and 42 is not divisible by 3+1, so 42 does not belong to the sequence.

Mathematica

Select[Range[20000], Function[n, AllTrue[Position[Reverse@ IntegerDigits[n, 2], 1][[All, 1]], Divisible[n, #] &]]] (* Michael De Vlieger, Dec 12 2022 *)

Prog

(PARI) is(n) = { my (r=n, k); while (r, r-=2^k=valuation(r, 2); if (n%(k+1), return (0); ); ); return (1); }
(Python)
def ok(n): return all(n%(k+1) == 0 or not n&(1<<k) for k in range(n.bit_length()))
print([m for m in range(20000) if ok(m)]) # Michael S. Branicky, Dec 07 2022
(Python)
from itertools import count, islice
def A358970_gen(startvalue=0): # generator of terms >= startvalue
    return filter(lambda n:not any(n%i for i, b in enumerate(bin(n)[:1:-1], 1) if b=='1'), count(max(startvalue, 0)))
A358970_list = list(islice(A358970_gen(), 20)) # Chai Wah Wu, Dec 12 2022

Crossrefs

Cf. A058891, A271410, A320673.

Sequence in context: A065392 A030457 A296300 * A224470 A168247 A229056

Adjacent sequences: A358967 A358968 A358969 * A358971 A358972 A358973

Keyword

nonn,base,easy

Author

Rémy Sigrist, Dec 07 2022

Status

approved