A296364 -id:A296364 - cp's OEIS frontend

A296349 Position where binary expansion of n starts in the binary Champernowne sequence A030190.

0, 1, 2, 4, 6, 9, 12, 15, 18, 22, 26, 30, 34, 38, 42, 46, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95, 100, 105, 110, 115, 120, 125, 130, 136, 142, 148, 154, 160, 166, 172, 178, 184, 190, 196, 202, 208, 214, 220, 226, 232, 238, 244, 250, 256, 262, 268, 274, 280, 286, 292

Offset: 0

N. J. A. Sloane, Dec 16 2017

			A030190 begins as follows (the bits are indexed starting at 0):
0,
1,
1, 0,
1, 1,
1, 0, 0,
1, 0, 1,
1, 1, 0,
1, 1, 1,
1, 0, 0, 0,
1, 0, 0, 1,
1, 0, 1, 0,
1, 0, 1, 1,
1, 1, 0, 0,
1, 1, 0, 1,
1, 1, 1, 0,
1, 1, 1, 1,
1, 0, 0, 0, 0,
1, 0, 0, 0, 1,
...
4 = 1,0,0 begins at the 6th bit, so a(4)=6; 5 = 1,0,1 begins at the 9th bit, so a(5)=9.

Cf. A030190, A030302, A296364.
This is A083652 prefixed by an initial 0.
A296354 is closely related.
Essentially partial sums of A070939.

A296356 a(n) = A296354(n) - A296355(n).

Original entry on oeis.org

0, 0, 5, 3, 21, 19, 23, 11, 65, 53, 59, 72, 74, 81, 70, 31, 169, 182, 166, 176, 183, 148, 202, 188, 210, 202, 180, 228, 218, 216, 185, 79, 441, 345, 411, 467, 433, 458, 416, 475, 449, 489, 436, 461, 516, 374, 509, 462, 538, 487, 537, 505, 522, 503, 577, 560

Offset: 0

N. J. A. Sloane, Dec 14 2017, corrected and extended Dec 17 2017

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

Rémy Sigrist, Table of n, a(n) for n = 0..16384

Cf. A076478, A296354, A295355, A116700, A296364.

More terms from Rémy Sigrist, Dec 19 2017

cp's OEIS Frontend

A296349 Position where binary expansion of n starts in the binary Champernowne sequence A030190.

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A296356 a(n) = A296354(n) - A296355(n).

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