A048991 -id:A048991 - cp's OEIS frontend

A263443 A self-describing sequence: when the sequence is read as a string of decimal digits, a(n) gives the starting position of an occurrence of n. This sequence is the lexicographically earliest one with this property.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 14, 1, 17, 130, 21, 50, 15, 28, 180, 33, 20, 37, 2, 200, 42, 52, 47, 270, 162, 60, 57, 310, 300, 3, 66, 350, 35, 73, 380, 78, 400, 41, 84, 302, 4, 91, 460, 96, 480, 22, 104, 510, 110, 530, 115, 5, 55, 122, 580, 53, 132, 146, 136

Offset: 1

Paul Tek, Oct 18 2015

The sequence does not necessarily give the earliest position of a number.

For example, 1234 first appears at position 1, but a(1234) = 28011.

			The following table lists few first terms, with the corresponding digits induced in the overall sequence:
+----+------+------------------------------------------------------------+
| n  | a(n) | New known digits                                           |
+----+------+------------------------------------------------------------+
|  1 |    1 | 1                                                          |
|  2 |    2 |  2                                                         |
|  3 |    3 |   3                                                        |
|  4 |    4 |    4                                                       |
|  5 |    5 |     5                                                      |
|  6 |    6 |      6                                                     |
|  7 |    7 |       7                                                    |
|  8 |    8 |        8                                                   |
|  9 |    9 |         9                                                  |
| 10 |   10 |          10                                                |
| 11 |   14 |            1411                                            |
| 12 |    1 |                                                            |
| 13 |   17 |                713                                         |
| 14 |  130 |                   0                                 ... 14 |
| 15 |   21 |                    215                                     |
| 16 |   50 |                       0                          16        |
| 17 |   15 |                        15                                  |
| 18 |   28 |                          2818                              |
+----+------+------------------------------------------------------------+

Paul Tek, Table of n, a(n) for n = 1..10000
Paul Tek, PERL program for this sequence

Cf. A048991, A114315, A125132, A210423.

Perl
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See Links section.
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A308540 a(n) is the least nonnegative number whose digits do not appear in order (not necessarily consecutively) in the concatenation of all previous terms.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 22, 33, 44, 55, 66, 77, 88, 99, 100, 332, 554, 776, 998, 1001, 3322, 5544, 7766, 9988, 33220, 33221, 55445, 77667, 99889, 332200, 332211, 776674, 776675, 3322008, 3322009, 7766741, 7766755, 33220088, 33220099, 77667411

Offset: 1

Rémy Sigrist, Jul 24 2019

This sequence is a variant of A048991.

			The first terms are necessarily the one-digit numbers: a(1) = 0, a(2) = 1, ..., a(10) = 9.
The number 10 does not appear in "0123456789", hence a(11) = 10.
The digits of every number from 11 to 21 appear in order in "012345678910", but this is not the case for the number 22, hence a(12) = 22.

Rémy Sigrist, C program for A308540

See A309340 for the binary variant.

Cf. A048991.

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A119615 Write down the primes 2,3,5,... but omit any prime (such as 23 or 31 or 71 ...) that has appeared as a string earlier in the sequence.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 29, 37, 41, 43, 47, 53, 59, 61, 67, 73, 79, 83, 89, 97, 101, 103, 107, 109, 127, 137, 139, 149, 151, 157, 163, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 277, 281, 283, 307, 311, 313, 331

Offset: 0

Gil Broussard, Jun 05 2006

Inspired by Hannah Rollman and her sequence A048991.

			23 is omitted since we see "2,3" at the beginning of the sequence; 15647 is omitted because 5641, 5647 appear earlier in the sequence.

Cf. A048991.

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<

A190784 Numbers whose binary representation is a substring of the concatenation of the binary representation of all smaller nonnegative integers not listed earlier, taken in decreasing order.

Original entry on oeis.org

2, 6, 7, 9, 11, 12, 14, 17, 19, 20, 21, 22, 25, 26, 27, 28, 29, 30, 31, 33, 34, 37, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 67, 69, 71, 72, 73, 74, 77, 78, 80, 81, 82, 84, 86, 87, 89, 90, 92, 93, 94

Offset: 1

M. F. Hasler, Dec 29 2012

Also, nonnegative integers which do not occur in A118250.

Up to the reversed (decreasing) order of concatenation, a binary analog of Hannah Rollman's numbers A048992.

			The binary representation of 2="10"[2] is a substring of the concatenation of 1 and 0, therefore a(1)=2. This term a(1)=2="10" will henceforth be excluded from the concatenations considered in the sequel.
The binary representations of 3, 4 and 5 are not a substrings of concat("1", "0") resp. concat("11", "1", "0") resp. concat("100", "11", "1", "0"). (Note that 2="10" is not among the concatenated numbers.)
But 6="110"[2] is again a substring of concat(5="101", 4="100", 3="11", "1", "0"), therefore a(2)=6. In the sequel, a(2)=6="110" will now also be always excluded from the concatenations, as is a(1)=2.

Analog of A128291 for the "with reversal" variant A118250 of A118248.

Cf. A048991, A048992.

A259236 Increasing sequence of numbers n such that the digits of n appear as a substring of the concatenation of the terms < n of the sequence, which is seeded with 1..9.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 23, 34, 45, 56, 67, 78, 89, 91, 99, 122, 123, 199, 212, 221, 222, 223, 231, 232, 233, 234, 312, 319, 322, 323, 332, 333, 334, 343, 344, 345, 431, 433, 434, 443, 444, 445, 454, 455, 456, 543, 544, 545, 554, 555, 556, 565, 566, 567, 654, 655, 656, 665, 666, 667, 676, 677, 678, 765, 766, 767, 776, 777

Offset: 1

Anthony Sand, Jun 22 2015

In A048991, n is excluded if it appears in the concatenation of all earlier terms. This sequence applies the opposite criterion and excludes n if it does NOT appear in the concatenation of all earlier terms. For example, the sequences starts with 1, 2, 3, ..., therefore 12, 23 and 123 appear in the sequence, but 10, 11 and 14 do not.

			The digits of 12 appear earlier in the sequence (1, 2...), therefore 12 is included.
The digits of 11 do not appear earlier in the sequence, therefore 11 is excluded.

Anthony Sand, Table of n, a(n) for n = 1..1000

Cf. A048991.

M:= 4: # to get all terms with <= M digits
with(StringTools):
S:= "123456789":
nS:= length(S):
Substrings:= {seq(seq(SubString(S,a..b),b=a+1..min(9,a+M-1)),a=1..8)}:
Cands:= map(parse,Substrings):
for n from 1 to 9 do A[n]:= n od:
for n from 10 while Cands <> {} do
  m:= min(Cands);
  A[n]:= m;
  S:= cat(S,convert(m,string));
  nm:= length(m);
  newSubstrings:= {seq(seq(SubString(S,a..b), b = a+nm-1..min(nS+nm, a+M-1)),a=1+nS-M .. nS)};
  Cands:= select(`>`,Cands union map(parse,newSubstrings), m);
  nS:= length(S);
od:
seq(A[i],i=1..n-1); # Robert Israel, Jun 22 2015

{ dmx=1000; d=vector(dmx); b=10; for(i=1,b-1,d[i]=i;print1(i,", ")); di=b-1; n=di; while(di

digits(n,i=1,j) = substring(sequence,i=1,j))

A322094 a(n) = number of occurrences of the decimal representation of n in the concatenation of all terms preceding a(n), or n if there are no such occurrences.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 1, 24, 25, 26, 27, 28, 29, 30, 1, 32, 33, 1, 35, 36, 37, 38, 39, 40, 1, 1, 43, 44, 1, 46, 47, 48, 49, 50, 1, 1, 1, 54, 55, 1, 57, 58, 59, 60, 1, 1, 1, 1, 65, 66, 1, 68, 69, 70, 1, 1

Offset: 1

John Mason, Nov 26 2018

In the definition of the sequence, consider the number of occurrences of string s in string t to be the number of positions within t that have a perfect match with the digits in s. Thus the number of occurrences of 11 in 1111 is 3 and not 2.

			a(12) is 1 as there is one occurrence of "12" in the string formed by concatenating a(1) through a(11).

John Mason, Table of n, a(n) for n = 1..10000

Cf. A048991, A048992.

FromDigits /@ Nest[Function[{a, n}, Append[a, If[# == 0, IntegerDigits@ n, IntegerDigits@ #] &@ SequenceCount[Join @@ a, IntegerDigits@ n]]] @@ {#, Length@ # + 1} &, {{1}}, 67] (* Michael De Vlieger, Nov 26 2018 *)

A103607 Write down the semiprimes but omit any semiprime (such as 46 or 69) that is the concatenation of consecutive semiprimes.

Original entry on oeis.org

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 49, 51, 55, 57, 58, 62, 65, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 121, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 169, 177, 178, 183, 185, 187

Offset: 1

Jonathan Vos Post, Jun 07 2006

The complement of this sequence is the sequence of semiprimes which are concatenations of successive semiprimes.

Note that this sequence is not analogous to A119615 for two reasons. In A119615 partial concatenation is taken into account, i.e., the terms 7, 11 prevent 71 to be included, while here only full concatenation is considered (hence 58, 62 do not forbid 86). Moreover in A119615 the terms to be concatenated are those in the sequence itself, while here are all the semiprimes. - Giovanni Resta, Jun 16 2016

			46 is not a term because concatenate(sp(1),sp(2)) = 46 = 2 * 23.
69 is not a term because concatenate(sp(2),sp(3)) = 69 = 3 * 23.
469 is not a term because concatenate(sp(1),sp(2),sp(3)) = 469 = 7 * 67.
1415 is not a term because concatenate(sp(5),sp(6)) = 1415 = 5 * 283.
2122 is not a term because concatenate(sp(7),sp(8)) = 2122 = 2 * 1061.
3839 is not a term because concatenate(sp(14),sp(15)) = 3839 = 11 * 349.
469101415 is not a term because concatenate(sp(1),sp(2),sp(3),sp(4),sp(5),sp(6)) = 469101415 = 5 * 93820283.
Where sp(i) is A001358(i).

Cf. A001358, A048991, A119615.

Name edited by Giovanni Resta, Jun 16 2016

A255724 List of numbers such that no number, nor its reverse, is in the concatenation of all previous terms.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 22, 24, 25, 26, 27, 29, 30, 33, 35, 36, 37, 38, 40, 44, 46, 47, 49, 50, 55, 57, 58, 60, 66, 69, 70, 77, 80, 88, 90, 99, 100, 102, 103, 104, 105, 106, 107, 108, 109, 112, 114, 115, 116, 117, 118, 122

Offset: 1

Phil Carmody, Mar 19 2015

			After listing 1..11, 12 is not listed as "12" is found in the concatenated earlier terms. After continuing with 13..18, 19 is not listed as "91" is likewise found.

Cf. A048991.

lista(nn) = my(d, v=[]); for(n=1, nn, for(i=0, #v-#d=digits(n), (v[i+1..i+#d]==d || v[i+1..i+#d]==Vecrev(d)) && next(2)); print1(n, ", "); v=concat(v, d)) \\ Jinyuan Wang, Aug 23 2021

$s="";$=0;do{$++;if(index($s,$)<0 && index($s,reverse)<0){print("$ ");$s.=$_}}while(1);

More terms from Jinyuan Wang, Aug 23 2021

cp's OEIS Frontend

A263443 A self-describing sequence: when the sequence is read as a string of decimal digits, a(n) gives the starting position of an occurrence of n. This sequence is the lexicographically earliest one with this property.

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Perl

A308540 a(n) is the least nonnegative number whose digits do not appear in order (not necessarily consecutively) in the concatenation of all previous terms.

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C

A119615 Write down the primes 2,3,5,... but omit any prime (such as 23 or 31 or 71 ...) that has appeared as a string earlier in the sequence.

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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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PARI

A190784 Numbers whose binary representation is a substring of the concatenation of the binary representation of all smaller nonnegative integers not listed earlier, taken in decreasing order.

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A259236 Increasing sequence of numbers n such that the digits of n appear as a substring of the concatenation of the terms < n of the sequence, which is seeded with 1..9.

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Maple

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Formula

A322094 a(n) = number of occurrences of the decimal representation of n in the concatenation of all terms preceding a(n), or n if there are no such occurrences.

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Mathematica

A103607 Write down the semiprimes but omit any semiprime (such as 46 or 69) that is the concatenation of consecutive semiprimes.

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Extensions

A255724 List of numbers such that no number, nor its reverse, is in the concatenation of all previous terms.

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