editing

approved

nonn,look,base,new

proposed

editing

editing

proposed

Rémy Sigrist, <a href="/A360296/b360296.txt">Table of n, a(n) for n = 1..8192</a>

approved

editing

proposed

approved

editing

proposed

allocated a(1) = 1, and for Rémy Sigristany n > 1, a(n) is the sum of the terms of the sequence at indices k < n whose binary digits appear in order but not necessarily as consecutive digits in the binary representation of n.

1, 1, 1, 2, 3, 3, 2, 4, 8, 11, 8, 8, 11, 8, 4, 8, 20, 34, 26, 34, 51, 40, 20, 20, 40, 51, 34, 26, 34, 20, 8, 16, 48, 96, 76, 118, 186, 152, 76, 96, 208, 281, 186, 152, 208, 124, 48, 48, 124, 208, 152, 186, 281, 208, 96, 76, 152, 186, 118, 76, 96, 48, 16, 32

1,4

This sequence is a variant of A165418.

<a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>

a(n) = Sum_{k = 1..A301977(n-1)} a(A301983(n, k)) for any n > 1.

a(2^k) = 2^(k-1) for any k > 0.

a(2^k-1) = 2^(k-2) for any k > 1.

a(n) >= A165418(n).

The first terms, alongside the corresponding k's, are:

n a(n) k's

-- ---- ------------------

1 1 N/A

2 1 {1}

3 1 {1}

4 2 {1, 2}

5 3 {1, 2, 3}

6 3 {1, 2, 3}

7 2 {1, 3}

8 4 {1, 2, 4}

9 8 {1, 2, 3, 4, 5}

10 11 {1, 2, 3, 4, 5, 6}

11 8 {1, 2, 3, 5, 7}

12 8 {1, 2, 3, 4, 6}

13 11 {1, 2, 3, 5, 6, 7}

14 8 {1, 2, 3, 6, 7}

15 4 {1, 3, 7}

16 8 {1, 2, 4, 8}

(PARI) { for (n=1, #a=vector(64), print1 (a[n]=if (n==1, 1, s = [1]; b = binary(n); for (k=2, #b, s = setunion(s, apply(v -> 2*v+b[k], s))); sum(k=1, #s-1, a[s[k]]); )", ")) }

Cf. A165418, A301983.

allocated

nonn,base

Rémy Sigrist, Feb 02 2023

approved

editing

allocated for Rémy Sigrist

allocated

approved