A356419 Inverse of A067576 considered as a permutation of the positive integers.

1, 2, 3, 4, 5, 8, 6, 7, 12, 17, 9, 23, 13, 18, 10, 11, 30, 38, 24, 47, 31, 39, 14, 57, 48, 58, 19, 69, 25, 32, 15, 16, 68, 80, 81, 93, 94, 108, 40, 107, 123, 139, 49, 156, 59, 70, 20, 122, 174, 193, 82, 213, 95, 109, 26, 234, 124, 140, 33, 157, 41, 50, 21, 22, 138, 155, 256

Offset

1,2

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Index entries for sequences that are permutations of the natural numbers

Formula

Let w(n) = A000120(n) be the Hamming weight of n, p(n) = A068076(n), then a(n) = binomial(w(n)+p(n)+1, 2) - p(n).

Example

A067576(12) = 9, so a(9) = 12.

Prog

(PARI) a(n)=my(w=hammingweight(n), p=sum(i=1, n-1, hammingweight(i)==w)); binomial(w+p+1, 2) - p
(Python)
from math import comb
def A356419(n):
    c, k = 0, 0
    for i, j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i, k)
    return comb(n.bit_count()+c+1, 2)-c # Chai Wah Wu, Mar 02 2023

Crossrefs

Cf. A067576, A000120, A068076, A263017, A067587.

Sequence in context: A191710 A194061 A272616 * A364775 A085176 A354125

Adjacent sequences: A356416 A356417 A356418 * A356420 A356421 A356422

Keyword

nonn,look,easy

Author

Jianing Song, Aug 06 2022

Status

approved