A098067 -id:A098067 - cp's OEIS frontend

A097487 Write the nonprime positive integers on labels in numerical order, forming an infinite sequence L. Now consider the succession of single digits of A000040 (prime numbers): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 3 7 4 1 4 3 4 7 5 3 ... (A033308). This sequence gives an arrangement L that produces the same succession of digits, subject to the constraint that the smallest unused label must be used that does not lead to a contradiction.

235, 711, 1, 3171, 9, 232, 93, 1374, 14, 34, 75, 35, 96, 16, 77, 1737, 98, 38, 99, 710, 110, 310, 71091, 1312, 713, 1137, 1391, 4, 91, 51, 15, 716, 316, 717, 3179, 18, 119, 11931, 97199, 21, 12, 2322, 72, 292, 33, 2392, 412, 512, 57, 26, 32, 6, 92, 712, 772, 8

Offset: 1

Eric Angelini, Sep 19 2004; corrected Sep 23 2004

This could be roughly rephrased like this: "Rewrite in the most economical way the prime numbers 'pattern' using only nonprime numbers. Do not use any nonprime twice."

			We must begin with 2,3,5,7,11,13,... and we cannot represent "2" with the label "2" or "23", so the next possibility is the label "235" (first available nonprime number in L).

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Eric Angelini, Jeux de suites, in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Cf. A068663, A097968, A098067, A098099.

f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[PrimeQ@ m, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Prime@ Range@ 200], 56] (* Michael De Vlieger, Nov 29 2015, Version 10.2 *)

A228595 Lexicographically earliest sequence of distinct positive integers such that the concatenation of the terms equals the concatenation of the positive integers, and no term appears in its natural position.

Original entry on oeis.org

12, 34, 56, 78, 910, 1, 11, 2, 131, 4, 151, 6, 171, 8, 1920, 212, 22, 3, 242, 5, 262, 7, 282, 9, 303, 13, 23, 33, 43, 53, 63, 73, 83, 940, 414, 24, 344, 454, 64, 74, 84, 950, 515, 25, 35, 45, 55, 65, 75, 85, 960, 616, 26, 36, 46, 566, 676, 86, 970, 717, 27

Offset: 1

Paul Tek, Aug 27 2013

Leading zeros are forbidden.

For any n>0, the concatenation of the (n-1) first terms never equals the concatenation of the (a(n)-1) first positive integers.

			The positive integers:
+-+-+-+-+-+-+-+-+-+---+---+---+---+---+---+---+---+---+
|1|2|3|4|5|6|7|8|9|1 0|1 1|1 2|1 3|1 4|1 5|1 6|1 7|1 8| ...
+-+-+-+-+-+-+-+-+-+---+---+---+---+---+---+---+---+---+
This sequence:
+---+---+---+---+-----+-+---+-+-----+-+-----+-+-----+-+
|1 2|3 4|5 6|7 8|9 1 0|1|1 1|2|1 3 1|4|1 5 1|6|1 7 1|8| ...
+---+---+---+---+-----+-+---+-+-----+-+-----+-+-----+-+

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

Cf. A033307, A064809, A098067, A197124.

Perl
```
See Link section.
```

A098103 Consider the succession of single digits of the primes (A000040): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 ... (A033308). This sequence is the lexicographically earliest derangement of A000040 that produces the same succession of digits.

Original entry on oeis.org

23, 5, 7, 11, 13, 17, 19, 2, 3, 293, 137, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 1371391491511, 571, 631, 67173179181191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283

Offset: 1

Eric Angelini, Sep 22 2004

Derangement here means a(n) != A000040(n) for all n.
Original name: "Write each prime number >0 on a single label. Put the labels in numerical order to form an infinite sequence L. Consider the succession of single digits of L: 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 3 7 4 1 4 3 4 7 5 3 5 9 6 1 6 7 7 1 7 3 7 9... (see A033308). The sequence S gives a rearrangement of the labels that reproduces the same succession of digits, subject to the constraints that a label of L cannot represent itself, and the smallest label must be used that does not lead to a contradiction."
This could be roughly rephrased like this: "Rewrite in the most economical way the 'prime numbers pattern' using only prime numbers, but rearranged. Do not use any prime more than once."
a(180) has over 1000 digits. - Danny Rorabaugh, Nov 29 2015

			We must begin with "2,3,5,7,11,..." and we cannot have the first term be 2, the first prime, so the smallest available prime is 23.

Danny Rorabaugh, Table of n, a(n) for n = 1..179
Eric Angelini, Sequence re-writing
Eric Angelini, Sequence re-writing [Cached copy, with permission]

For other sequences of this type, cf. A098067.

f[lst_List, k_] := Block[{L = lst, g, a = {}, m = 0}, g[] := {Set[m, First@ FromDigits@ Append[IntegerDigits@ m, First@ #]], Set[L, Last@ #]} &@ TakeDrop[L, 1]; Do[g[]; While[Or[m == Prime[Length@ a + 1], ! PrimeQ@ m, MemberQ[a, m]], g[]]; AppendTo[a, m]; m = 0, {k}]; a]; f[Flatten@ Map[IntegerDigits, Prime@ Range@ 120], 53] (* Michael De Vlieger, Nov 29 2015, Version 10.2 *)

def A098103(n):
  Pr, p, s, A, i = Primes(), 2, "", [], 1
  while len(A)A098103(179) # Danny Rorabaugh, Nov 29 2015

Name, Comments, and Example edited by Danny Rorabaugh, Nov 28 2015

Corrected and extended by Danny Rorabaugh, Nov 29 2015

A097484 Write the odd positive integers on labels in numerical order, forming an infinite sequence L. Consider the succession of single digits of L: 1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 2 7 2 9 3 1 ... (A031312). This sequence is a derangement of L that produces the same succession of digits, subject to the constraint that the smallest unused label must be used that does not lead to a contradiction.

Original entry on oeis.org

13, 5, 7, 9, 1, 113, 15, 17, 19, 21, 23, 25, 27, 29, 3, 133, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 105, 107, 109, 11, 1113, 115, 117, 119, 121, 123, 125, 127, 129, 131

Offset: 1

Eric Angelini, Sep 19 2004

Derangement here means the n-th element of L is not the n-th element of this sequence, so a(n) != 2n - 1.

			We must begin with 1,3,5,7... and we cannot have a(1) = 1, so the next possibility is the label "13". The next term must be the smallest available label not leading to a contradiction, thus "5". The next one will be "7", etc. After the label "9" the smallest available label is "1". After this "1" we cannot have a(6) = 11 -- we thus take the smallest available label which is "113". No label is allowed to start with a leading zero.

Cf. A005408, A097481, A097485, A097487, A097488, A097500, A097912, A097962, A098067, A098080.

Same type of sequence -- but for even numbers -- is A097481. - Eric Angelini, Aug 12 2008

Corrected and extended by Jacques ALARDET and Eric Angelini, Aug 12 2008

Derangement wording introduced by Danny Rorabaugh, Nov 26 2015

A097485 Write the positive integers on labels in numerical order, forming an infinite sequence L. Consider now the succession of single digits made by juxtaposing Fibonacci numbers: 1 1 2 3 5 8 1 3 2 1 3 4 5 5 ... (A031324). This sequence gives a derangement of L that produces the same succession of digits, subject to the constraint that the smallest unused label must be used that does not lead to a contradiction.

Original entry on oeis.org

11, 23, 58, 1, 3, 2, 13, 4, 5, 589, 14, 42, 33, 37, 7, 6, 10, 9, 8, 71, 59, 72, 584, 41, 81, 67, 65, 109, 46, 17, 71, 12, 86, 57, 463, 68, 750, 25, 121, 39, 31, 96, 418, 317, 81, 151, 422, 98, 320

Offset: 1

Eric Angelini, Sep 19 2004

Labels of L can be used only once in this sequence.
We could name this sequence the "Fibo_nat_cci" sequence (nat stands for "natural numbers").
Derangement here means the n-th term of L is not the n-th term of the sequence, so a(n) != n.

			We must begin with 1,1,2,3,... and we cannot have a(1) = 1, so the next possibility is the label "11". After "68" we must get "7,5,0,2,5,1,2,1,3,9,3,1,9,6,4,1,8..." (corresponding to Fibonacci numbers "75025,121393,196418..."); "7" is already used, and we cannot use "75" since no label begins with a 0. So the next term is "750".

Cf. A000045, A031324, A098067.

Derangement wording introduced by Danny Rorabaugh, Nov 27 2015

Initial 0 removed by Danny Rorabaugh, Nov 28 2015

cp's OEIS Frontend

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A228595 Lexicographically earliest sequence of distinct positive integers such that the concatenation of the terms equals the concatenation of the positive integers, and no term appears in its natural position.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Perl

A098103 Consider the succession of single digits of the primes (A000040): 2 3 5 7 1 1 1 3 1 7 1 9 2 3 2 9 3 1 ... (A033308). This sequence is the lexicographically earliest derangement of A000040 that produces the same succession of digits.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Sage

Extensions

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions