cp's OEIS Frontend

a(n) = number of k such that A116700(k) <= n; a(n) = n - number of k such that A131881(k) <= n.

A131982 gives numbers n such that a(n) = n/2, or numbers n such that (number of k such that A116700(k) <= n) = (number of k such that A131881(k) <= n).

This is the binary "early-birdness" of n (cf. A116700, A296364).

Theorem: a(n) > 0 for all n > 1.

Proof. The claim is true for 2 <= n <= 7, so assume n >= 8, and let u = 1... denote the binary expansion of n. Let L denote the list of all binary vectors whose concatenation gives A076478.

To show a(n)>0 it is enough to exhibit a pair of successive binary vectors b, c in L whose concatenation contains a copy of u that begins in b and is such that b appears in L before u does. There are three cases.

(i) Suppose n is even, say u = 1x0. Take c = x00, and let b be the vector preceding c in L, so that b = y11, say. Then bc = y11x00 contains u.

(ii) Suppose n = 2^k-1, u = 1^k. Take b = 01^(k-1), c = 10^(k-1), so that bc = 0 1^k 0^(k-1).

(iii) Otherwise, n is an odd number whose binary expansion contains a 0, say u = 1^k 0x1. Take c = 0x10^k, and let b be the vector preceding c in L, so that b = y1^k, say, and bc = y1^k 0x10^k.

In each case we need to verify that b does appear in L before u, but we leave this easy verification to the reader. QED

N-th Early bird number A116700(n) is in the sequence if it occurs in the concatenation of the first n-1 early bird numbers, A116700(1), ..., A116700(n-1).

With A116700 as early bird numbers of order 1, this can be generalized to define early bird numbers of order k for k > 1: N-th Early bird number of order k-1 is an early bird number of order k if it occurs in the concatenation of the first n-1 early bird numbers of order k-1.

Inspired by Eric Angelini's posting to Seq Fan mailing list, Jul 23 2007.

N-th Early bird number of order 2 is in the sequence if it occurs in the concatenation of the first n-1 early bird numbers of order 2.

Every nonnegative integer appears finitely many times in this sequence.

Numbers n such that number of terms <= n of A116700 equals number of terms <= n of A131881.

Numbers n such that numbers of numbers that occur in the concatenation of 1,2,3...,n-1 equals numbers of numbers that do not occur in the concatenation of 1,2,3...,n-1.

There are no other terms <= 600000. The plots in the link strongly suggest that the sequence is complete.

See A133650 for definition of early bird numbers of order n.

a(28) > 6*10^7. [From Donovan Johnson, Nov 06 2010]

A161373 U {a(n)} = A000027.

Whether or not 22 is punctual or early bird is a matter interpretation of "early occurrence" in the definition of A161373: 10110 occurs as the right 3 bits of 21 (10101) and the left 2 bits of 22 (10110) itself, which is ahead of the natural position, but not *completely* ahead of it. One can show (see weblink) the 22 is the only such case of doubt. [From Hagen von Eitzen, Jun 29 2009]

Another measure of the binary "early-birdness" of n (cf. A296356, A116700).