A167500 -id:A167500 - cp's OEIS frontend

A167502 Sequence A167500 (= list of nonzero bits in this sequence), written in binary.

1, 10, 100, 111, 1000, 1001, 1010, 1110, 10001, 10010, 10100, 10110, 10111, 11000, 11010, 11110, 11111, 100010, 100100, 100110, 101001, 101011, 101100, 101110, 110000, 110001, 110010, 110011, 110100, 111000, 111001, 111011, 111101, 111110

Offset: 1

M. F. Hasler, Nov 05 2009

			Sequence A167500 starts 1,2,4,7,... = 1,10,100,111,.... when written in binary.

Cf. A167500 (this sequence written in decimal), A167501 (the digits of this sequence).

a=b=[]; for(n=1,30, #b >= n & for( i=a[n-1]+1,#b, b[i] & (a=concat(a,i)) & break); #asubst(Pol(binary(t)),x,10),a)

A167502 = A007088 o A167500

A167501 Sequence of bits in A167500 (= list of nonzero bits in this sequence, when written in binary).

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 0

Offset: 1

M. F. Hasler, Nov 05 2009

Sequence A167500 starts 1,2,4,7,... = 1,10,100,111,.... when written in binary (A167502). This sequence lists the bits one by one, MSB first: 1,1,0,1,0,0,1,1,1,...; by definition A167500 lists the nonzero terms of this function, and reciprocally this sequence is the characteristic function of A167500.

Cf. A167500 (lists nonzero terms in this sequence), A167502.

a=b=[]; for(n=1,30, #b >= n & for( i=a[n-1]+1,#b, b[i] & (a=concat(a,i)) & break); #a

A167522 Positive integers not occurring in A167500.

Original entry on oeis.org

3, 5, 6, 11, 12, 13, 15, 16, 19, 21, 25, 27, 28, 29, 32, 33, 35, 37, 39, 40, 42, 45, 47, 53, 54, 55, 58, 60, 65, 72, 73, 74, 76, 78, 79, 81, 82, 84, 85, 88, 90, 92, 93, 96, 98, 102, 105, 106, 108, 112, 115, 116, 117, 118, 121, 122, 123, 127, 128, 130, 133, 134, 139, 141

Offset: 1

M. F. Hasler, Nov 05 2009

Equivalently, positions of zero digits in A167502 (when all terms are concatenated).

Cf. A167500-A167503, A167520-A167523.

{a=b=[]; for(n=1,99, #b>=n & for(i=a[n-1]+1,#b,b[i] & (a=concat(a,i)) & break); #a

A167519 Lexicographically earliest increasing sequence which lists the positions of the zero digits in the sequence.

Original entry on oeis.org

3, 10, 11, 12, 11000, 11111, 11112, 11113, 11114, 11115, 11116, 11117, 11118, 11119, 11121, 11122, 11123, 11124, 11125, 11126, 11127, 11128, 11129, 11131, 11132, 11133, 11134, 11135, 11136, 11137, 11138, 11139, 11141, 11142, 11143, 11144

Offset: 1

M. F. Hasler, Nov 05 2009

The terms of the sequence give the positions of the digits '0' in the string formed by concatenating all the terms (written in base 10).

			The sequence cannot start with 1 (which would mean it starts with 0) or 2 (which would mean that the second term equals 0), so a(1)=3 is the smallest possibility.
Thereafter, the smallest possible value for a(2), which must have '0' as second digit, is a(2)=10.
This means that the next digit '0' must occur at position 10; up to there, we use the smallest possible values for a(3)=11 and a(4)=12.
Then must follow two nonzero digits (which must be part of a(5)) and then three zero digits (from a(2),a(3),a(4) = 10, 11, 12). None of the latter can be the first digit of a(6), so they must be part of a(5), for which the smallest possibility is therefore a(5)=11000.
This also means that there is no digit '0' between the 12th digit (= the last digit of a(6)), and the 11000th digit of the sequence. So there follow roughly 11000/5 terms which are the smallest possible 5-digit terms without a zero digit.

Cf. A167500-A167503. See A210414 for another version.

Edited by Charles R Greathouse IV, Apr 24 2010

Definition corrected by Jaroslav Krizek, Jun 19 2014

A167520 Positions of nonzero digits in this sequence, when all terms are concatenated.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 73, 74, 75, 76

Offset: 1

M. F. Hasler, Nov 05 2009

Numbers are written in base 10 in the usual way; the terms give the positions of the nonzero digits when all terms are concatenated.

The sequence starts at index 1 since it is a list. The sequence cannot start with a(1)=0, this would mean that the first nonzero digit occurs at position 0, which does not make sense, or at best is a contradiction. Thus we cannot number the positions starting from zero, and the first digit (most significant digit of the first term) is necessarily nonzero, i.e. a(1)=1.

			The term which follows a(1)=1 (cf. comment) will be nonzero and thus have a nonzero most significant digit, which is the digit that comes right after the initial 1, thus digit # 2 is nonzero, therefore a(2)=2.
This goes on the like up to a(10)=10 (this refers to the most significant digit of the 10th term, which is necessarily nonzero). But the first digit of this "10" is followed by the digit '0', so the next nonzero digit is not at position 11, but it will come in the following position, a(11)=12.
From here on, there is always an increasing number of subsequent digits already written down, and it is sufficient to list the positions of the digits different from '0'.

Cf. A167500 and A167502 (analog for base 2).

base(n,b=10) = { local( a=[ n%b ]); while( 0= n & for( i=a[n-1]+1,#b, b[i] & (a=concat(a,i)) & break); #a

A167503 Positions of nonzero digits in this sequence, where the terms are written in base 3 (and concatenated).

Original entry on oeis.org

1, 2, 10, 12, 20, 21, 100, 101, 102, 112, 121, 122, 201, 202, 210, 211, 212, 220, 221, 222, 1000, 1001, 1002, 1011, 1012, 1021, 1022, 1100, 1102, 1110, 1111, 1112, 1120, 1121, 1122, 1200, 1202, 1210, 1211, 1212, 1220, 1221, 1222, 2010, 2020, 2021, 2101

Offset: 1

M. F. Hasler, Nov 05 2009

This sequence lists the positions of nonzero digits in the sequence, where the terms are written in base 3 and concatenated, such that the least significant digit (units) of a(n) is followed by the most significant digit of a(n+1).

			The list cannot start with 0 (cf. A167500), so the first digit of the sequence is nonzero, whence a(1)=1.
The next nonzero digit of this sequence will be the most significant digit (m.s.d.) of a(2), necessarily nonzero, thus a(2)=2.
For the same reason, a(3) = 3[10] = 10[3]. (All terms are written in base 3.)
Thus the 4th digit is zero, followed by the m.s.d. of the next term which is nonzero, thus a(4)=5.
Terms of the sequence:_ 1,2,10,12,20,21,100,101
Position of the digits: 1 2 34 56 78 9A BCD EFG (A=10,...)
Thus the numbers 4,8,12,13,15,..., which give the positions of digits '0', are not in this sequence.

base(n,b=3,s=1) = { my( a=[ n%b ]); while( 0=n & for(i=a[n-1]+1,#b,b[i] & (a=concat(a,i)) & break); #a

Edited by Charles R Greathouse IV, Aug 02 2010

A377862 A variant of Golomb's sequence (A001462): the n-th digit of the sequence gives the number of times n appears, with a(1) = 1 and a(2) = 2.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13, 13, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 19, 19

Offset: 1

Rémy Sigrist, Nov 10 2024

This sequence is a base-10 variant of A167500.
Numbers corresponding to positions of zeros do not appear in the sequence.
This sequence first differ from A001462 for n = 169: a(169) = 31 whereas A001462(169) = 29.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Scatterplot of the first 500 of this sequence and of A001462
Rémy Sigrist, PARI program

Cf. A001462, A087739, A167500, A377863 (missing numbers), A377896.

PARI
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A167523 Positions of zero digits in A167503 (when all terms are concatenated).

Original entry on oeis.org

4, 8, 12, 13, 15, 18, 30, 33, 37, 46, 54, 55, 56, 58, 59, 62, 63, 66, 70, 74, 78, 83, 84, 87, 92, 104, 115, 116, 119, 124, 136, 146, 148, 150, 152, 154, 159, 163, 176, 183, 184, 187, 192, 210, 211, 212, 213, 215, 216, 217, 220, 221, 225, 226, 230, 231, 235, 236, 240

Offset: 1

M. F. Hasler, Nov 05 2009

Also, positive integers not occurring in A167503 (bearing in mind that the terms of A167503 are written in base 3, but here numbers are written in base 10).

			Sequence A167503: 1,2,10,12,20,21,100,101
Digit's position: 1 2 34 56 78 9A BCD EFG (A=10,...)
Thus the digits '0' occur at positions 4,8,12,13,15,...: this sequence.

Cf. A167500-A167503.

base(n,b=3,s=1) = { my( a=[ n%b ]); while( 0=n && for(i=a[n-1]+1,#b,b[i] && (a=concat(a,i)) && break); #a

Edited by Charles R Greathouse IV, Aug 02 2010

A287515 a(n) = position of n-th 0 when sequence is written in base 2.

Original entry on oeis.org

2, 7, 8, 9, 11, 12, 15, 20, 21, 27, 29, 30, 32, 34, 38, 44, 50, 52, 53, 54, 55, 56, 58, 59, 60, 62, 64, 65, 68, 70, 73, 74, 77, 78, 80, 83, 85, 86, 89, 91, 95, 98, 101, 108, 109, 110, 114, 116, 120, 127, 128, 134, 136, 137, 138, 139, 140, 141, 143, 144, 145, 146, 147, 150, 151, 152, 154, 155, 157, 158, 159, 162

Offset: 1

Anthony Sand, May 26 2017

A167500 lists the positions of 1's when the sequence is written in binary. This sequence lists the positions of 0's. When written in binary, it begins 10, 111, 1000, 1001, 1011... The first 0 appears at position 2, so a(1) = 2 = 10. The second 0 appears at position 7, so a(2) = 7 = 111. The third 0 appears at position 8, so a(3) = 8 = 1000. The sequence then becomes self-generating, because entries are added to it faster than 0's are detected in it.

			a(1) = zeropos([10...],1) = 2,
a(2) = zeropos([10,111,1000...],2) = 7,
a(3) = zeropos([10,111,1000...],3) = 8,
a(4) = zeropos([10,111,1000...],4) = 9,
a(5) = zeropos([10,111,1000,1001...],5) = 11.

Anthony Sand, Table of n, a(n) for n = 1..10000

Cf. A167500, A167502.

{ zeroposseq()= smx=100; s=vector(smx); s[1]=2; s[2]=7; s[3]=8; si=0; dig=digits(s[1],2); di=1; i=1; dl=0; while(si#dig, di++; dig=digits(s[di],2); i=1; ); ); }

a(n) = zeropos([sequence],n).

A167521 Positive integers not occurring in A167520.

Original entry on oeis.org

11, 29, 47, 67, 85, 105, 123, 143, 161, 181, 182, 184, 187, 190, 193, 196, 199, 202, 205, 209, 239, 266, 296, 323, 353, 380, 410, 445, 446, 448, 451, 454, 457, 460, 463, 467, 497, 527, 554, 584, 614, 641, 671, 701, 727, 728, 730, 733, 736, 739, 742, 745, 748

Offset: 1

M. F. Hasler, Nov 05 2009

Equivalently, positions of zero digits in A167520 (when all terms are concatenated).

The first differences are (18, 18, 20, 18, 20, 18, 20, 18, 20, 1, 2, 3, 3, 3, 3, 3, 3, 3, 4, 30, 27, 30, 27, 30, 27, 30, 35, 1, 2, 3, 3, 3, 3, 3, 4, ...)

			The first occurrence of the digit '0' in A167520 is as the least significant digit of A167520(10)=10, which occurs at position 11, thus a(1)=11. Equivalently, this is the least positive integer missing in A167520.
The next occurrence of the digit '0' in A167520 is at position 29 (and 29 is the second positive integer not occurring in A167520), thus a(2)=29.

Cf. A167500-A167503, A167520-A167523.

base(n,b=10) = { my( a=[ n%b ]); while( 0=n & for(i=a[n-1]+1,#b,b[i] & (a=concat(a,i)) & break); #a

cp's OEIS Frontend

A167502 Sequence A167500 (= list of nonzero bits in this sequence), written in binary.

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A167501 Sequence of bits in A167500 (= list of nonzero bits in this sequence, when written in binary).

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A167522 Positive integers not occurring in A167500.

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A167519 Lexicographically earliest increasing sequence which lists the positions of the zero digits in the sequence.

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A167520 Positions of nonzero digits in this sequence, when all terms are concatenated.

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A167503 Positions of nonzero digits in this sequence, where the terms are written in base 3 (and concatenated).

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A377862 A variant of Golomb's sequence (A001462): the n-th digit of the sequence gives the number of times n appears, with a(1) = 1 and a(2) = 2.

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A167523 Positions of zero digits in A167503 (when all terms are concatenated).

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A287515 a(n) = position of n-th 0 when sequence is written in base 2.

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