A355625 a(1) = 1; for n > 1, a(n) is the number of terms in the first n-1 terms of the sequence that share a 1-bit with n in their binary expansions.

1, 0, 1, 0, 2, 1, 4, 0, 3, 2, 6, 2, 6, 7, 11, 0, 6, 9, 13, 6, 13, 13, 18, 6, 11, 17, 21, 16, 21, 22, 26, 0, 14, 16, 26, 14, 23, 25, 31, 12, 22, 27, 34, 27, 33, 34, 39, 19, 31, 35, 43, 36, 44, 44, 49, 36, 42, 48, 52, 47, 52, 53, 57, 0, 29, 32, 48, 30, 48, 48, 57, 25, 41, 46, 56, 47, 57, 58, 65, 34

Offset: 1

Author

Scott R. Shannon, Jul 10 2022

Keywords

• nonn
• base

Comments

The indices where a(n) = 1 in the first 500000 terms are 1, 3, 6. It is likely no more exist although this is unknown. Many terms of the sequence are close to the line a(n) = n although only the first term is a possible fixed point. In the first 500000 terms the lowest values not to appear are 5, 8, 10, 15, 20, 24, 28. It is likely these and other numbers never appear although this is unknown. All terms for n > 1 where n is a power of 2 equal 0.

Examples

			a(7) = 4 as the total number of terms in the first six terms that share a 1-bit with 7 in their binary expansions is four, namely 1, 1, 2, 1.
		

Links

• Scott R. Shannon, Table of n, a(n) for n = 1..10000
• Scott R. Shannon, Image of the first 500000 terms. The green line is y = n.

Crossrefs

Cf. A355621, A030190, A129760, A353989, A352763, A354606.

Programs

Python

from itertools import count, islice
def agen():
    an, alst = 1, [1]
    for n in count(2):
        yield an
        an = sum(1 for k in alst if k&n)
        alst.append(an)
print(list(islice(agen(), 80))) # Michael S. Branicky, Jul 10 2022