A359080 Numbers k such that A246600(k) = A000005(k).

1, 3, 5, 7, 11, 13, 15, 17, 19, 23, 27, 29, 31, 37, 41, 43, 47, 51, 53, 59, 61, 63, 67, 71, 73, 79, 83, 85, 89, 95, 97, 101, 103, 107, 109, 111, 113, 119, 123, 125, 127, 131, 137, 139, 143, 149, 151, 157, 163, 167, 173, 179, 181, 187, 191, 193, 197, 199, 211, 219

Offset: 1

Author

Amiram Eldar, Dec 15 2022

Keywords

• nonn
• base

Comments

Numbers k such that for all the divisors d of k the bitwise OR of k and d is equal to k (or equivalently, the bitwise AND of k and d is equal to d).

Subsequence of A102553. Terms of A102553 that are not in this sequence: 2, 135, 175, 243, 343, ... .

All the terms are odd since if k is even and d = 1 then bitor(k, 1) > k and thus A246600(k) < A000005(k).

All the odd primes are terms.

All the numbers of the form 2^k-1 (A000225) are terms.

Numbers k such that the bitwise OR(k, d_1, d_2, ..., d_m) = k, where d_1, ..., d_m are the divisors of k. - Chai Wah Wu, Dec 18 2022

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to binary expansion of n.
• Index entries for sequences related to divisors of numbers.

Crossrefs

Subsequence of A102553.

Subsequences: A000225, A065091.

Cf. A000005, A246600, A359081, A359082, A359083.

Programs

Mathematica

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; Select[Range[250], s[#] == DivisorSigma[0, #] &]

PARI

is(n) = sumdiv(n, d, bitor(d, n) == n) == numdiv(n);

Python

from itertools import count, islice
from operator import ior
from functools import reduce
from sympy import divisors
def A359080_gen(startvalue=1):  # generator of terms >= startvalue
    return filter(
        lambda n: n | reduce(ior, divisors(n, generator=True)) == n,
        count(max(startvalue, 1)),
    )
A359080_list = list(islice(A359080_gen(), 20))  # Chai Wah Wu, Dec 18 2022
print(A359080_list)