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

Author

Rémy Sigrist, Dec 07 2022

Keywords

• nonn
• base
• easy

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.

Examples

			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.
		

Links

• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A058891, A271410, A320673.

Programs

Mathematica

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

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<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