A161373 -id:A161373 - cp's OEIS frontend

A161374 "Punctual" binary numbers. Complement of A161373.

0, 1, 2, 4, 8, 10, 16, 22, 32, 36, 64, 128, 136, 256, 512, 528, 1024, 2048, 2080, 4096, 8192, 8256, 16384, 32768, 32896, 65536, 131072, 131328, 262144, 524288, 524800, 1048576, 2097152, 2098176, 4194304, 8388608, 8390656, 16777216, 33554432

Offset: 1

Paolo P. Lava and Giorgio Balzarotti, Jun 08 2009

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]

H. v. Eitzen, Binary Early Birds (2009). [From _Hagen von Eitzen_, Jun 29 2009]
Index entries for linear recurrences with constant coefficients, signature (0,0,6,0,0,-8).

Cf. A083655, A116700, A161373.

From Hagen von Eitzen, Jun 29 2009: (Start)
G.f.: (1+x+2x^2)/(2-8x^3) + x/(2-4x^3) -1/2 -x + x^4 + 4x^5 + 2x^6 + 6x^7 + 6x^8
If q>=3 then a(3q) = 2^(2q-1), a(3q+1) = 2^(2q-1) + 2^(q-1), a(3q+2) = 2^(2q). (End)
a(n) = A083655(n-2) for n>=9. - Alois P. Heinz, Dec 14 2022

Offset corrected as customary for lists, 20 removed by Hagen von Eitzen, Jun 27 2009

More terms from Hagen von Eitzen, Jun 29 2009

A118248 Least nonnegative integer whose binary representation does not occur in the concatenation of the binary representations of all earlier terms.

Original entry on oeis.org

0, 1, 2, 4, 7, 8, 11, 16, 18, 21, 22, 25, 29, 31, 32, 35, 36, 38, 40, 58, 64, 67, 68, 70, 75, 76, 78, 87, 88, 90, 93, 99, 101, 104, 107, 122, 128, 131, 133, 134, 136, 138, 140, 144, 148, 150, 152, 155, 156, 159, 161, 169, 170, 172, 178, 183, 188, 190

Offset: 0

Leroy Quet, Apr 18 2006

Otherwise said: Omit numbers whose binary representation already occurs in the concatenation of the binary representation of earlier terms. As such, a binary analog of A048991 / A048992 (Hannah Rollman's numbers), rather than "early bird" binary numbers A161373. - M. F. Hasler, Jan 03 2013

Rainer Rosenthal, Table of n, a(n) for n = 0..9999
Nick Hobson, Python program for this sequence

Cf. A118247 (concatenation of binary representations), A118250, A118252 (variants where binary representations are reversed).

Block[{b = {{0}}, a = {0}, k, d}, Do[k = FromDigits[#, 2] &@ Last@ b + 1; While[SequenceCount[Flatten@ b, Set[d, IntegerDigits[k, 2]]] > 0, k++]; AppendTo[b, d]; AppendTo[a, k], {i, 57}]; a] (* Michael De Vlieger, Aug 19 2017 *)

A118248(n,show=0,a=0)={my(c=[a],find(t,s,L)=L || L=#s; for(i=0,#t-L, vecextract( t,(2^L-1)<M. F. Hasler, Jan 03 2013

Perl

$s="";$i=0;do{$i++;$b=sprintf("%b",$i);if(index($s,$b)<0){print("$i=$b\n");$s.=$b}}while(1);

Extensions

More terms from Graeme McRae, Apr 19 2006

Explicit definition from M. F. Hasler, Dec 29 2012

Perl program by Phil Carmody, Mar 19 2015

Crossref and Perl program by Phil Carmody, Mar 19 2015

A296365 Numbers which appear prematurely in the binary Champernowne word (A030190).

Original entry on oeis.org

3, 5, 6, 7, 9, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85

Offset: 1

N. J. A. Sloane, Dec 17 2017

This is the complement of A083655.

Cf. A030190, A161373.

A333920 a(n) is the least k such that the binary representation of n appears as a substring in the concatenation of the binary representations of 0, 1, ..., k.

Original entry on oeis.org

0, 1, 2, 2, 4, 3, 2, 4, 8, 5, 10, 3, 4, 3, 4, 8, 16, 9, 5, 10, 19, 11, 22, 4, 8, 5, 10, 3, 4, 7, 8, 16, 32, 17, 9, 18, 36, 5, 10, 20, 35, 19, 11, 11, 38, 22, 4, 8, 16, 9, 5, 13, 20, 11, 22, 4, 8, 5, 20, 7, 8, 15, 16, 32, 64, 33, 17, 34, 9, 35, 18, 36, 69, 37

Offset: 0

Rémy Sigrist, Apr 10 2020

Every nonnegative integer appears finitely many times in this sequence.

Rémy Sigrist, Table of n, a(n) for n = 0..8192

Cf. A047778, A161373, A161374, A333921 (decimal variant).

a(n, base=2) = { my (w=base^#digits(n, base), m=0); for (k=0, oo, my (d=if (k, digits(k, base), [0])); for (i=1, #d, m=(base*m+d[i])%w; if (m==n, return (k)))) }

a(n) <= n with equality iff n belongs to A161374.

a(A047778(n)) = n for any n > 0.

A187752 Number of times the binary representation of n occurs in the concatenation of the binary representation of all smaller numbers.

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 2, 0, 1, 0, 3, 2, 3, 4, 3, 0, 1, 1, 2, 1, 2, 0, 6, 2, 3, 3, 5, 5, 4, 6, 4, 0, 1, 1, 2, 0, 3, 2, 3, 1, 3, 1, 4, 1, 3, 3, 8, 2, 3, 4, 4, 3, 5, 3, 8, 5, 5, 5, 6, 8, 5, 8, 5, 0, 1, 1, 2, 1, 2, 2, 3, 1, 2, 2, 4, 1, 5, 3, 4, 1, 3, 2, 5, 2, 4, 2, 6, 1, 4, 3, 6, 2, 6, 4, 10, 2, 3, 4, 4, 3

Offset: 0

M. F. Hasler, Jan 03 2013

nonn

Related to "early bird" (decimal: A116700, binary: A161373) and Hannah Rollman's numbers (cf. A048991, A048992 for decimal; A118248 and A118247-A118251 for binary versions). The latter would correspond to a variant of this sequence which has indices of nonzero terms omitted from the concatenation.

			a(3) = 1 since concatenation of 0,1,2 in binary yields "0110", and 3 = "11"[2] occurs once in this string.

(nMax)->my(c=[],cnt(t,s,M)=M=2^#s-1;sum(i=0,#t-#s,vecextract(t,M<

A229123 a(n) gives the number of bases, b>1, in which n is an early bird.

Original entry on oeis.org

0, 0, 1, 0, 2, 2, 3, 2, 3, 2, 4, 3, 6, 4, 5, 3, 7, 2, 7, 5, 7, 6, 7, 4, 9, 7, 6, 5, 8, 5, 10, 4, 8, 8, 7, 5, 13, 8, 8, 6, 12, 7, 12, 7, 8, 11, 11, 5, 13, 9, 12, 9, 11, 5, 13, 11, 13, 12, 12, 5, 17, 11, 11, 8, 13, 9, 14, 9, 12, 7, 14, 8, 18, 11, 9, 11, 13, 11

Offset: 1

Paul Tek, Sep 14 2013

A number n is called an early bird in base b, if its digits in base b appear in the concatenation of the digits in base b of the numbers from 1 to n-1.

			The number 1 is never an early bird, so a(1)=0.
The number 3 is an early bird only in base 2, so a(3)=1.
The number 7 is an early bird in bases 2, 3 and 5, so a(7)=3.

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, C program for this sequence
Paul Tek, Illustration of the bases in which n is an early bird, where n ranges from 1 to 1000

Cf. A116700, A161373, A135549.

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A161374 "Punctual" binary numbers. Complement of A161373.

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A118248 Least nonnegative integer whose binary representation does not occur in the concatenation of the binary representations of all earlier terms.

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A296365 Numbers which appear prematurely in the binary Champernowne word (A030190).

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A333920 a(n) is the least k such that the binary representation of n appears as a substring in the concatenation of the binary representations of 0, 1, ..., k.

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A187752 Number of times the binary representation of n occurs in the concatenation of the binary representation of all smaller numbers.

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A229123 a(n) gives the number of bases, b>1, in which n is an early bird.

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