cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A301807 Lexicographically first sequence of distinct integers whose concatenation of digits is the same as the concatenation of the digits of the absolute differences between consecutive terms.

Original entry on oeis.org

1, 2, 4, 8, 16, 15, 9, 10, 5, 14, 24, 19, 18, 22, 20, 61, 52, 34, 12, 32, 26, 11, 13, 47, 35, 3, 29, 28, 17, 51, 44, 41, 36, 33, 31, 40, 38, 30, 205, 191, 147, 134, 71, 68, 37, 77, 39, 69, 49, 54, 53, 62, 63, 64, 60, 67, 66, 100, 93, 92, 86, 78, 75, 82, 89, 96, 57, 126, 122, 27, 23, 76, 70, 72, 135, 129, 125, 65, 59, 825
Offset: 1

Views

Author

Eric Angelini and Jean-Marc Falcoz, Mar 27 2018

Keywords

Comments

This sequence might not be a permutation of A000027 (the positive numbers). After 18000 terms the smallest integer not yet present is 42. This 42 will perhaps never show.
From Rémy Sigrist, Jul 04 2018: (Start)
In fact, a(18420) = 42; however that this sequence is a permutation of the natural numbers remains an open question.
If we drop the unicity constraint, then we obtain A210025.
If moreover we impose that the sequence be nondecreasing, then we obtain A100787.
(End)

Examples

			(The first members of the equalities hereunder must be seen as absolute differences between the successive pairs of adjacent terms:)
    1 -  2 =  1
    2 -  4 =  2
    4 -  8 =  4
    8 - 16 =  8
   16 - 15 =  1
   15 -  9 =  6
    9 - 10 =  1
   10 -  5 =  5
    5 - 14 =  9
   14 - 24 = 10
   24 - 19 =  5
   19 - 18 =  1, etc.
We see that the first and the last column present the same digit succession: 1, 2, 4, 8, 1, 6, 1, 5, 9, 1, 0, 5, 1, ...

Links

Crossrefs

Cf. A301743 for the same idea with additions of adjacent terms instead of absolute differences.
Cf. A100787, A210025.

A365257 The five digits of a(n) and their four successive absolute first differences are all distinct.

Original entry on oeis.org

14928, 15829, 17958, 18259, 18694, 18695, 19372, 19375, 19627, 25917, 27391, 27398, 28149, 28749, 28947, 34928, 35917, 37289, 37916, 38926, 39157, 39578, 43829, 45829, 47289, 47916, 49318, 49681, 49687, 51869, 53719, 57391, 57398, 58926, 59318, 59681, 59687, 61973, 61974, 62983, 62985, 67958, 68149, 68749, 68947, 69157, 69578, 71952, 71953, 72691, 72698, 74619, 74982, 74986, 75193, 75196, 76859, 78259, 78694, 78695, 81394, 81395, 81539, 82941, 82943, 85179, 85629, 85971, 85976, 86749, 87269, 87593, 87596, 89372, 89375, 89627, 91647, 91735, 92658, 92834, 92851, 92854, 93518, 94182, 94186, 94768, 94782, 94786, 95281, 95287, 95867, 96278, 96815, 97158, 98273, 98274
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, Aug 29 2023

Keywords

Comments

The digit 0 is never present in a(n) and never appears as a first difference (as this would duplicate in both cases one of the 8 remaining digits involved).
The sequence ends with a(96) = 98274.
The only prime numbers with this property are 39157, 49681, 51869, 53719, 62983, 68749, 68947, 75193, 78259, 89627 and 95287.

Examples

			The five digits of a(1) = 14928 produce the four successive absolute first differences 3 (= 1 - 4), 5 (= 4 - 9), 7 (= 9 - 2) and 6 (= 2 - 8), resulting in nine distinct digits.
.1.4.9.2.8.
..3.5.7.6..

Crossrefs

Cf. A365258, A100787, A040114, A270263.

Programs

  • Mathematica
    Select[Range[10000,99999],Sort@Join[IntegerDigits@#, Abs@Differences@IntegerDigits@#]==Range@9&]

A365258 The four digits of a(n), their three successive absolute first differences and their two successive absolute second differences are all distinct.

Original entry on oeis.org

2983, 3892, 4197, 4917, 5298, 5928, 7194, 7398, 7914, 7938, 8139, 8295, 8329, 8397, 8925, 8937, 9238, 9318
Offset: 1

Views

Author

Eric Angelini and Giorgos Kalogeropoulos, Aug 29 2023

Keywords

Comments

The digit 0 is never present in a(n) and never appears as a first or a second difference (as this would duplicate in both cases one of the 8 remaining digits involved).
The sequence ends with a(18) = 9318.

Examples

			2983 is a term since its three successive absolute first differences 7 (= 2 - 9), 1 (= 9 - 8), 5 (= 8 - 3) and the successive absolute second differences 6 (= 7 - 1) and 4 (= 1 - 5), are nine distinct digits.
  2 9 8 3
   7 1 5
    6 4

Crossrefs

Cf. A365257, A100787, A040114, A270263.

Programs

  • Mathematica
    Select[Range[1000,9999],Sort@Join[IntegerDigits@#, s=Abs@Differences@IntegerDigits@#, Abs@Differences@s]==Range@9&]

A362987 Lexicographically earliest sequence S of distinct positive terms such that the successive digits of S are the successive spreads of S' terms (see Comments for definition of "spread").

Original entry on oeis.org

10, 11, 12, 21, 23, 13, 20, 32, 24, 14, 34, 25, 31, 22, 30, 35, 42, 15, 43, 26, 36, 37, 46, 16, 41, 45, 53, 57, 47, 33, 52, 27, 40, 64, 54, 38, 48, 58, 68, 17, 63, 28, 69, 18, 51, 39, 56, 60, 59, 65, 62, 49, 50, 74, 61, 29, 73, 70, 85, 96, 72, 75, 79, 81, 84, 44, 71, 95, 83, 105, 104, 19
Offset: 1

Views

Author

Eric Angelini, May 12 2023

Keywords

Comments

The spread of n is the absolute difference between the leftmost digit of n and the rightmost digit of n. Spreads vary from 0 to 9.

Examples

			a(1) = 10 with spread 1;
a(2) = 11 with spread 0;
a(3) = 12 with spread 1;
a(4) = 21 with spread 1;
a(5) = 23 with spread 1;
a(6) = 13 with spread 2; etc.
We see that the above succession of spreads is the digits' succession of S.

Crossrefs

Cf. A100787.

Programs

  • Mathematica
    a[1]=10;a[n_]:=a[n]=Block[{k=10},While[Abs[First@#-Last@#]&@IntegerDigits[k][[{1,-1}]]!=Flatten[IntegerDigits/@Array[a,n-1]][[n]]||MemberQ[Array[a,n-1],k],k++];k];Array[a,72] (* Giorgos Kalogeropoulos, May 12 2023 *)
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