A355621 - OEIS

A355621

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 a(n-1) in their binary expansions.

1, 1, 2, 1, 3, 5, 5, 6, 5, 8, 1, 8, 2, 4, 5, 11, 15, 17, 12, 11, 19, 17, 15, 23, 22, 19, 22, 21, 24, 16, 10, 18, 20, 21, 29, 33, 22, 30, 33, 23, 38, 31, 42, 28, 35, 37, 38, 37, 40, 22, 41, 40, 24, 33, 35, 46, 49, 49, 50, 47, 59, 60, 55, 61, 62, 61, 64, 1, 39, 63, 69, 58, 60, 64, 3, 60, 65, 46, 67

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OFFSET

1,3

COMMENTS

The indices where a(n) = 1 in the first 500000 terms are 1, 2, 4, 11, 68, 131, 2051, 4099. It is unknown if more exist. 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 7, 9, 14, 25, 26. It is likely these and other numbers never appear although this is unknown.

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.

EXAMPLE

a(7) = 5 as a(6) = 5 and the total number of terms in the first six terms that share a 1-bit with 5 in their binary expansions is five, namely 1, 1, 1, 3, 5.

PROG

(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&an)

alst.append(an)

print(list(islice(agen(), 79))) # Michael S. Branicky, Jul 10 2022

CROSSREFS

Cf. A355625, A030190, A129760, A353989, A352763, A354606.