A031312 -id:A031312 - cp's OEIS frontend

A098099 Consider the succession of single digits of the positive odd integers: 1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 ... (A031312). This sequence is the lexicographically earliest sequence of distinct positive even integers that produces the same succession of digits.

1357911131517192, 12, 32, 52, 72, 931333537394, 14, 34, 54, 74, 951535557596, 16, 36, 56, 76, 971737577798, 18, 38, 58, 78, 9919395979910, 110, 310, 510, 710, 911111311511711912, 112, 312, 512, 712, 913113313513713914, 114, 314, 514, 714

Offset: 1

Eric Angelini, Sep 22 2004

Original name: "Write each even integer >0 on a single label. Put the labels in numerical order to form an infinite sequence L. Now consider the succession of single digits of A005408 (odd numbers): 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 3 3 3 5 3 7 3 9... The sequence S gives a rearrangement of the labels that reproduces the same succession of digits, subject to the constraint that 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 "odd numbers pattern" using only even numbers, but rearranged. All the numbers of the sequence must be different one from another.

			We must begin with "1,3,5..." and we cannot use "1" or "13" or "135" (only even terms are available), so the first possibility is "1357911131517192". For "199,201,203..." we won't be allowed to use "1992", for instance, since no term begins with a 0.

E. Angelini, "Jeux de suites", in Dossier Pour La Science, pp. 32-35, Volume 59 (Jeux math'), April/June 2008, Paris.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A005408, A005843, A097487, A097968.

f[lst_List, k_] := Block[{L = lst, g, w, a = {}, m}, g[x_] := First@ FirstPosition[x, i_ /; EvenQ@ i]; Do[w = Take[L, g@ L]; L = Drop[L, Length@ w]; m = Take[L, g@ L]; While[Or[MemberQ[a, FromDigits@ w], IntegerLength@ FromDigits@ m < Length@ m], w = Join[w, m]; L = Drop[L, Length@ m]; m = Take[L, g@ L]]; AppendTo[a, FromDigits@ w], {k}]; a]; f[Flatten@ Map[IntegerDigits, Range[1, 1000, 2]], 35] (* Michael De Vlieger, Nov 28 2015, Version 10 *)

Name and Example edited by Danny Rorabaugh, Nov 28 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

A031313 Position of n-th 0 in A031312.

Original entry on oeis.org

97, 100, 103, 106, 109, 247, 250, 253, 256, 259, 397, 400, 403, 406, 409, 547, 550, 553, 556, 559, 697, 700, 703, 706, 709, 847, 850, 853, 856, 859, 997

Offset: 1

Clark Kimberling

nonn

A031314 Position of n-th 1 in A031312.

Original entry on oeis.org

1, 6, 7, 8, 10, 12, 14, 17, 27, 37, 47, 57, 67, 77, 87, 96, 98, 99, 102, 105, 108, 111, 112, 113, 114, 115, 117, 118, 120, 121, 123, 124, 126, 128, 129, 132, 135, 138, 141, 143, 144, 147, 150, 153, 156, 158, 159, 162, 165, 168, 171

Offset: 1

Clark Kimberling

nonn

A031315 Position of n-th 2 in A031312.

Original entry on oeis.org

16, 18, 20, 22, 24, 127, 130, 133, 136, 139, 246, 249, 252, 255, 258, 261, 264, 267, 270, 273, 276, 277, 279, 280, 282, 283, 285, 286, 288, 289, 291, 294, 297, 300, 303, 306, 309, 312, 315, 318, 321, 324, 327, 330, 333, 336, 339

Offset: 1

Clark Kimberling

nonn

A031316 Position of n-th 3 in A031312.

Original entry on oeis.org

2, 9, 19, 26, 28, 29, 30, 32, 34, 39, 49, 59, 69, 79, 89, 101, 116, 131, 142, 145, 146, 148, 151, 154, 161, 176, 191, 206, 221, 236, 251, 266, 281, 292, 295, 296, 298, 301, 304, 311, 326, 341, 356, 371, 386, 396, 399, 401, 402, 405

Offset: 1

Clark Kimberling

nonn

A031317 Position of n-th 4 in A031312.

Original entry on oeis.org

36, 38, 40, 42, 44, 157, 160, 163, 166, 169, 307, 310, 313, 316, 319, 457, 460, 463, 466, 469, 546, 549, 552, 555, 558, 561, 564, 567, 570, 573, 576, 579, 582, 585, 588, 591, 594, 597, 600, 603, 606, 607, 609, 610, 612, 613, 615

Offset: 1

Clark Kimberling

nonn

Cf. A031312.

Flatten[Position[Flatten[IntegerDigits[2*Range[500]-1]],4]] (* Harvey P. Dale, Aug 12 2015 *)

A031318 Position of n-th 5 in A031312.

Original entry on oeis.org

3, 11, 21, 31, 41, 46, 48, 50, 51, 52, 54, 61, 71, 81, 91, 104, 119, 134, 149, 164, 172, 175, 178, 179, 181, 184, 194, 209, 224, 239, 254, 269, 284, 299, 314, 322, 325, 328, 329, 331, 334, 344, 359, 374, 389, 404, 419, 434, 449

Offset: 1

Clark Kimberling

nonn

A031319 Position of n-th 6 in A031312.

Original entry on oeis.org

56, 58, 60, 62, 64, 187, 190, 193, 196, 199, 337, 340, 343, 346, 349, 487, 490, 493, 496, 499, 637, 640, 643, 646, 649, 787, 790, 793, 796, 799, 846, 849, 852, 855, 858, 861, 864, 867, 870, 873, 876, 879, 882, 885, 888, 891, 894

Offset: 1

Clark Kimberling

nonn

A031320 Position of n-th 7 in A031312.

Original entry on oeis.org

4, 13, 23, 33, 43, 53, 63, 66, 68, 70, 72, 73, 74, 83, 93, 107, 122, 137, 152, 167, 182, 197, 202, 205, 208, 211, 212, 214, 227, 242, 257, 272, 287, 302, 317, 332, 347, 352, 355, 358, 361, 362, 364, 377, 392, 407, 422, 437, 452

Offset: 1

Clark Kimberling

nonn

cp's OEIS Frontend

A098099 Consider the succession of single digits of the positive odd integers: 1 3 5 7 9 1 1 1 3 1 5 1 7 1 9 2 1 2 3 2 5 ... (A031312). This sequence is the lexicographically earliest sequence of distinct positive even integers that produces the same succession of digits.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Mathematica

Extensions

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions

A031313 Position of n-th 0 in A031312.

Views

Author

Keywords

A031314 Position of n-th 1 in A031312.

Views

Author

Keywords

A031315 Position of n-th 2 in A031312.

Views

Author

Keywords

A031316 Position of n-th 3 in A031312.

Views

Author

Keywords

A031317 Position of n-th 4 in A031312.

Views

Author

Keywords

Crossrefs

Programs

Mathematica

A031318 Position of n-th 5 in A031312.

Views

Author

Keywords

A031319 Position of n-th 6 in A031312.

Views

Author

Keywords

A031320 Position of n-th 7 in A031312.

Views

Author

Keywords