A363690 Numbers k such that A246600(k) = 2.

3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 56, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 76, 77, 79, 80, 81, 82, 83, 84, 86, 88

Offset: 1

Author

Amiram Eldar, Jun 16 2023

Keywords

• nonn
• base
• easy

Comments

A subsequence of A080943 and first differs from it at n = 42: A080943(42) = 55 is not a term of this sequence.

Numbers k such that A246600(k) = 1 are the powers of 2 (A000079).

Numbers k that have exactly 2 divisors d such that the bitwise AND of k and d is equal to d, or equivalently, the bitwise OR of k and d is equal to k. These two divisors are k and the highest power of 2 dividing k, A006519(k).

Includes all the even squarefree semiprimes (i.e., the odd primes doubled, A100484 \ {4}).

If k is a term then 2*k is also a term. The primitive terms are the odd terms of this sequence, A363691.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 6, 76, 681, 6268, 60002, 587247, 5811449, 57817051, 576781821, 5761341533, 57583082392, 575687822743, ... . Apparently, the asymptotic density of this sequence exists and equals 0.575... .

Links

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

Crossrefs

Cf. A000079, A006519, A080943, A100484, A246600, A363691.

Programs

Mathematica

q[n_] := DivisorSum[n, Boole[BitOr[#, n] == n] &] == 2; Select[Range[100], q]

PARI

is(n) = sumdiv(n, d, bitor(d, n) == n) == 2;

Python

from itertools import count, islice
from sympy import divisors
def A363690_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda n:(m:=n&-n)!=n and all(d==m or d==n or n|d!=n for d in divisors(n,generator=True)),count(max(startvalue,2)))
A363690_list = list(islice(A363690_gen(),20)) # Chai Wah Wu, Jun 20 2023