A233272 a(n) = n + 1 + number of nonleading zeros in binary representation of n (A080791).

1, 2, 4, 4, 7, 7, 8, 8, 12, 12, 13, 13, 15, 15, 16, 16, 21, 21, 22, 22, 24, 24, 25, 25, 28, 28, 29, 29, 31, 31, 32, 32, 38, 38, 39, 39, 41, 41, 42, 42, 45, 45, 46, 46, 48, 48, 49, 49, 53, 53, 54, 54, 56, 56, 57, 57, 60, 60, 61, 61, 63, 63, 64, 64, 71, 71, 72

Offset: 0

Author

Antti Karttunen, Dec 12 2013

Keywords

• nonn
• base

Comments

From Antti Karttunen, Jan 30 2022: (Start)

Write n in binary: 1ab..xyz, then a(n) = (1+1ab..xy) + (1+1ab..x) + ... + (1+1ab) + (1+1a) + (1+1) + (1+0) + 1. This method was found by LODA miner, see the assembly program at C. Krause link.

Proof: Compare to a similar formula given for A011371, with a(n) = a(floor(n/2)) + floor(n/2) to the new formula for this sequence which is a(n) = 1 + a(floor(n/2)) + floor(n/2), for n > 0 and a(0) = 1. It is easy to see that the difference between these, a(n) - A011371(n) = 1+A070939(n), for n > 0. As A011371(n) = n minus (number of 1's in binary expansion of n), then a(n) = 1 + (number of digits in binary expansion of n) + (n minus number of 1's in binary expansion of n) = 1 + n + (number of nonleading 0's in binary expansion of n), which indeed is the definition of this sequence.

(End)

Links

• Antti Karttunen, Table of n, a(n) for n = 0..8192
• Christian Krause, et al, A mined LODA assembly source for this sequence
• Index entries for sequences related to binary expansion of n

Crossrefs

Bisection: A233273 (A120511 + 1). Iterates: A233271, A216431.

Cf. A000120, A005187, A011371, A054429, A070939, A080791, A092391.

Programs

Mathematica

DigitCount[#, 2, 0] + # + 1 & [Range[0, 100]] (* Paolo Xausa, Mar 01 2024 *)

PARI

A233272(n) = { my(s=1); while(n, n>>=1; s+=(1+n)); (s); }; \\ (After a LODA-assembly program found by a miner) - Antti Karttunen, Jan 30 2022

Scheme

(define (A233272 n) (+ 1 n (A080791 n)))
;; Alternatively:
(define (A233272 n) (if (zero? n) 1 (+ n (A000120 (A054429 n)))))

Formula

a(n) = n + A080791(n) + 1.

For all n>=1, a(n) = n + A000120(A054429(n)).

a(0) = 1; for n > 1, a(n) = 1 + floor(n/2) + a(floor(n/2)). - (Found by LODA miner, see comments) - Antti Karttunen, Jan 30 2022