OFFSET
1,3
COMMENTS
This sequence is a variant of A295277: here we count earlier terms with common one bits, there without common one bits.
The scatterplot of this sequence shares features with that of A295277 (see scatterplot in Links section).
Empirically, this sequence and A295277 have tight connections:
- let c be the sequence defined for any n > 0 by c(n) = a(n) + A295277(n),
- we have for any n > 0, c(n) <= c(n+1) <= c(n) + 1,
- let u be the sequence defined for any n > 0 by u(n) = least k such that a(k) = n,
- let v be the sequence defined for any n >= 0 by v(n) = least k such that A295277(k) = n,
- we have for any n > 0, u(n) + 1 = v(n).
LINKS
Rémy Sigrist, Table of n, a(n) for n = 1..25000
Rémy Sigrist, Scatterplot of the first 2^20 terms
Rémy Sigrist, Colored scatterplot of the first 2^20 terms (where the color is function of min(A000120(a(n)),A000120((Max_{k=1..n-1} a(k))+2-a(n))))
Rémy Sigrist, Scatterplot of (n, a(n) + A295277(n)) for n=1..2^20
FORMULA
a(2^n) = 0 for any n >= 0.
EXAMPLE
The first terms, alongside the distinct earlier terms with common one bits with n, are:
n a(n) Distinct earlier terms with common one bits with n
-- ---- --------------------------------------------------
1 1 {}
2 1 {}
3 2 {1}
4 1 {}
5 2 {1}
6 2 {2}
7 3 {1, 2}
8 1 {}
9 3 {1, 3}
10 3 {2, 3}
11 4 {1, 2, 3}
12 2 {4}
13 4 {1, 3, 4}
14 4 {2, 3, 4}
15 5 {1, 2, 3, 4}
16 1 {}
17 4 {1, 3, 5}
18 3 {2, 3}
19 5 {1, 2, 3, 5}
20 3 {4, 5}
PROG
(PARI) mx = 0; for (n=1, 85, v = 1 + sum(i=1, mx, bitand(i, n)!=0); print1 (v ", "); mx = max(mx, v))
KEYWORD
nonn,base
AUTHOR
Rémy Sigrist, Nov 19 2017
STATUS
approved