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.

Previous Showing 31-34 of 34 results.

A271991 g_n(10) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

10, 29, 65, 125, 215, 284, 363, 452, 551, 660, 779, 907, 1045, 1193, 1351, 1519, 1697, 1885, 2083, 2291, 2509, 2737, 2975, 3222, 3479, 3746, 4023, 4310, 4607, 4914, 5231, 5558, 5895, 6242, 6599, 6966, 7343, 7730, 8127, 8534, 8951
Offset: 0

Views

Author

Natan Arie Consigli, May 22 2016

Keywords

Comments

For more info see A266201-A266202.

Crossrefs

Cf. A271557: G_n(10).
Weak Goodstein sequences: A137411: g_n(11); A265034: g_n(266); A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A271988: g_n(7); A271989: g_n(8); A271990: g_n(9); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0;

Programs

  • Mathematica
    g[k_, n_] := If[k == 0, n, Total@ Flatten@ MapIndexed[#1 (k + 2)^(#2 - 1) &, Reverse@ IntegerDigits[#, k + 1]] &@ g[k - 1, n] - 1]; Table[g[n, 10], {n, 0, 40}]

A271992 g_n(16) where g is the weak Goodstein function defined in A266202.

Original entry on oeis.org

16, 80, 169, 310, 515, 795, 1163, 1631, 2211, 2915, 3755, 4742, 5889, 7208, 8711, 10410, 12317, 14444, 16803, 19406, 22265, 25392, 28799, 32472, 36447, 40736, 45351, 50304, 55607, 61272, 67311, 73736, 80559, 87792, 95447, 103536, 112071
Offset: 0

Views

Author

Natan Arie Consigli, May 24 2016

Keywords

Comments

For more information see A266201 and A266202.

Crossrefs

Cf. A271557: G_n(10).
Weak Goodstein sequences: A267647: g_n(4); A267648: g_n(5); A271987: g_n(6); A271988: g_n(7); A271989: g_n(8); A271990: g_n(9); A271991: g_n(10); A137411: g_n(11); A265034: g_n(266); A266202: g_n(n); A266203: a(n)=k such that g_k(n)=0.

Programs

  • Mathematica
    g[k_, n_] :=
If[k == 0, n,
  Total@Flatten@
       MapIndexed[#1 (k + 2)^(#2 - 1) &,
        Reverse@IntegerDigits[#, k + 1]] &@g[k - 1, n] - 1]; Table[
g[n, 16], {n, 0, 36}]

A361838 a(n) is the number of 2s in the binary hereditary representation of 2n.

Original entry on oeis.org

1, 2, 3, 2, 3, 4, 5, 3, 4, 5, 6, 5, 6, 7, 8, 3, 4, 5, 6, 5, 6, 7, 8, 6, 7, 8, 9, 8, 9, 10, 11, 4, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 9, 10, 11, 12, 7, 8, 9, 10, 9, 10, 11, 12, 10, 11, 12, 13, 12, 13, 14, 15, 4, 5, 6, 7, 6, 7, 8, 9, 7, 8, 9, 10, 9, 10, 11, 12, 7
Offset: 1

Views

Author

Jodi Spitz, Mar 26 2023

Keywords

Comments

See comments on A266201 for the definition of hereditary representation.

Examples

			A table of n, the binary hereditary representation of 2n, and the number of 2s in the representation:
 n | hereditary rep. of 2n   | number of 2s
---+-------------------------+--------------
 1 | 2                       |      1
 2 | 2^2                     |      2
 3 | 2^2+2                   |      3
 4 | 2^(2+1)                 |      2
 5 | 2^(2+1)+2               |      3
 6 | 2^(2+1)+2^2             |      4
 7 | 2^(2+1)+2^2+2           |      5
 8 | 2^2^2                   |      3
 9 | 2^2^2+2                 |      4
10 | 2^2^2+2^2               |      5
11 | 2^2^2+2^2+2             |      6
12 | 2^2^2+2^(2+1)           |      5
13 | 2^2^2+2^(2+1)+2         |      6
14 | 2^2^2+2^(2+1)+2^2       |      7
15 | 2^2^2+2^(2+1)+2^2+2     |      8
16 | 2^(2^2+1)               |      3
17 | 2^(2^2+1)+2             |      4
18 | 2^(2^2+1)+2^2           |      5
19 | 2^(2^2+1)+2^2+2         |      6
20 | 2^(2^2+1)+2^(2+1)       |      5
21 | 2^(2^2+1)+2^(2+1)+2     |      6
22 | 2^(2^2+1)+2^(2+1)+2^2   |      7
23 | 2^(2^2+1)+2^(2+1)+2^2+2 |      8
24 | 2^(2^2+1)+2^2^2         |      6
25 | 2^(2^2+1)+2^2^2+2       |      7
26 | 2^(2^2+1)+2^2^2+2^2     |      8
27 | 2^(2^2+1)+2^2^2+2^2+2   |      9
28 | 2^(2^2+1)+2^2^2+2^(2+1) |      8

Links

Crossrefs

Cf. A005245, A025280.

Programs

  • PARI
    a(n)=if(n==0, 0, sum(k=0, logint(n,2), if(bittest(n,k), 1 + a((k+1)\2)))) \\ Andrew Howroyd, Apr 07 2023

A296441 Array A(n, k) = G_k(n) where G_k(n) is the k-th term of the Goodstein sequence of n, read by antidiagonals.

Original entry on oeis.org

0, 0, 1, 0, 0, 2, 0, 0, 2, 3, 0, 0, 1, 3, 4, 0, 0, 0, 3, 26, 5, 0, 0, 0, 2, 41, 27, 6, 0, 0, 0, 1, 60, 255, 29, 7, 0, 0, 0, 0, 83, 467, 257, 30, 8, 0, 0, 0, 0, 109, 775, 3125, 259, 80, 9, 0, 0, 0, 0, 139, 1197, 46655, 3127, 553, 81, 10, 0, 0, 0, 0, 173, 1751, 98039, 46657, 6310, 1023, 83, 11
Offset: 0

Views

Author

Iain Fox, Dec 12 2017

Keywords

Comments

G_0(n) = n. To get to the second term in the row, convert n to hereditary base 2 representation (see links), replace each 2 with a 3, and subtract 1. For the third term, convert the second term (G_1(n)) into hereditary base 3 notation, replace each 3 with a 4, and subtract one. This pattern continues until the sequence converges to 0, which, by Goodstein's Theorem, occurs for all n.

Examples

			| n\k |  0   1    2     3      4      5       6       7       8       9  ...
|-----|------------------------------------------------------------------------
|  0  |  0,  0,   0,    0,     0,     0,      0,      0,      0,      0, ...
|  1  |  1,  0,   0,    0,     0,     0,      0,      0,      0,      0, ...
|  2  |  2,  2,   1,    0,     0,     0,      0,      0,      0,      0, ...
|  3  |  3,  3,   3,    2,     1,     0,      0,      0,      0,      0, ...
|  4  |  4, 26,  41,   60,    83,   109,    139,    173,    211,    253, ...
|  5  |  5, 27, 255,  467,   775,  1197,   1751,   2454,   3325,   4382, ...
|  6  |  6, 29, 257, 3125, 46655, 98039, 187243, 332147, 555551, 885775, ...
| ... |

Links

Crossrefs

n-th row: A000004 (n=0), A000007 (n=1), A215409 (n=3), A056193 (n=4), A266204 (n=5), A266205 (n=6), A271554 (n=7), A271555 (n=8), A271556 (n=9), A271557 (n=10), A271558 (n=11), A271559 (n=12), A271560 (n=13), A271561 (n=14), A222117 (n=15), A059933 (n=16), A271562 (n=17), A271975 (n=18) A211378 (n=19), A271976 (n=20).
k-th column: A001477 (k=0), A056004 (k=1), A057650 (k=2), A059934 (k=3), A059935 (k=4), A059936 (k=5), A271977 (k=6), A271978 (k=7), A271979 (k=8), A271985 (k=9), A271986 (k=10).
G_n(n) = A266201(n) (main diagonal of array).

Programs

  • PARI
    B(n, b)=sum(i=1, #n=digits(n, b), n[i]*(b+1)^if(#n
Previous Showing 31-34 of 34 results.

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