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.

Showing 1-3 of 3 results.

A257997 Numbers of the form (2^i)*(3^j) or (2^i)*(5^j) or (3^i)*(5^j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 27, 32, 36, 40, 45, 48, 50, 54, 64, 72, 75, 80, 81, 96, 100, 108, 125, 128, 135, 144, 160, 162, 192, 200, 216, 225, 243, 250, 256, 288, 320, 324, 375, 384, 400, 405, 432, 486, 500, 512, 576, 625, 640
Offset: 1

Views

Author

Reinhard Zumkeller, May 16 2015

Keywords

Comments

Union of A003586, A003592 and A003593.
Subsequence of 5-smooth numbers (cf. A051037), having no more than two distinct prime factors: A006530(a(n)) <= 5; A001221(a(n)) <= 2.

Examples

			. ----+------+---------     ----+------+-----------
.   1 |   1  |  1            16 |  25  |  5^2
.   2 |   2  |  2            17 |  27  |  3^3
.   3 |   3  |  3            18 |  32  |  2^5
.   4 |   4  |  2^2          19 |  36  |  2^2 * 3^2
.   5 |   5  |  5            20 |  40  |  2^3 * 5
.   6 |   6  |  2 * 3        21 |  45  |  3^2 * 5
.   7 |   8  |  2^3          22 |  48  |  2^4 * 3
.   8 |   9  |  3^2          23 |  50  |  2 * 5^2
.   9 |  10  |  2 * 5        24 |  54  |  2 * 3^3
.  10 |  12  |  2^2 * 3      25 |  64  |  2^6
.  11 |  15  |  3 * 5        26 |  72  |  2^3 * 3^2
.  12 |  16  |  2^4          27 |  75  |  3 * 5^2
.  13 |  18  |  2 * 3^2      28 |  80  |  2^4 * 5
.  14 |  20  |  2^2 * 5      29 |  81  |  3^4
.  15 |  24  |  2^3 * 3      30 |  96  |  2^5 * 3

Links

Crossrefs

Cf. A003586, A003592, A003593, A051037, A006530, A001221, A258023 (subsequence).
Cf. A337800, A337801.

Programs

  • Haskell
    import Data.List.Ordered (unionAll)
a257997 n = a257997_list !! (n-1)
a257997_list = unionAll [a003586_list, a003592_list, a003593_list]
  • Mathematica
    n = 1000; Join[Table[2^i*3^j, {i, 0, Log[2, n]}, {j, 0, Log[3, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}], Table[2^i*5^j, {i, 0, Log[2, n]}, {j, 0, Log[5, n/2^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

Formula

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n/log(30))). - Vaclav Kotesovec, Sep 22 2020
Sum_{n>=1} 1/a(n) = 29/8. - Amiram Eldar, Sep 23 2020

A258023 Numbers of form (2^i)*(3^j) or (3^i)*(5^j).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 12, 15, 16, 18, 24, 25, 27, 32, 36, 45, 48, 54, 64, 72, 75, 81, 96, 108, 125, 128, 135, 144, 162, 192, 216, 225, 243, 256, 288, 324, 375, 384, 405, 432, 486, 512, 576, 625, 648, 675, 729, 768, 864, 972, 1024, 1125, 1152, 1215, 1296
Offset: 1

Views

Author

Reinhard Zumkeller, May 16 2015

Keywords

Comments

Union of A003586 and A003593;
A006530(a(n)) <= 5; A001221(a(n)) <= 2; a(n) mod 10 != 0.

Examples

			.   n |  a(n) |                 n |  a(n) |
. ----+-------+----------     ----+-------+------------
.   1 |    1  |  1             16 |   32  |  2^5
.   2 |    2  |  2             17 |   36  |  2^2 * 3^2
.   3 |    3  |  3             18 |   45  |  3^2 * 5
.   4 |    4  |  2^2           19 |   48  |  2^4 * 3
.   5 |    5  |  5             20 |   54  |  2 * 3^3
.   6 |    6  |  2 * 3         21 |   64  |  2^6
.   7 |    8  |  2^3           22 |   72  |  2^3 * 3^2
.   8 |    9  |  3^2           23 |   75  |  3 * 5^2
.   9 |   12  |  2^2 * 3       24 |   81  |  3^4
.  10 |   15  |  3 * 5         25 |   96  |  2^5 * 3
.  11 |   16  |  2^4           26 |  108  |  2^2 * 3^3
.  12 |   18  |  2 * 3^2       27 |  125  |  5^3
.  13 |   24  |  2^3 * 3       28 |  128  |  2^7
.  14 |   25  |  5^2           29 |  135  |  3^3 * 5
.  15 |   27  |  3^3           30 |  144  |  2^4 * 3^2

Links

Crossrefs

Cf. A003586, A003593, A051037, A006530, A001221, A010879, subsequence of: A051037, A257997, A337800, A337801.

Programs

  • Haskell
    import Data.List.Ordered (union)
a258023 n = a258023_list !! (n-1)
a258023_list = union a003586_list a003593_list
  • Mathematica
    n = 10^4; Join[Table[2^i*3^j, {i, 0, Log[2, n]}, {j, 0, Log[3, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

Formula

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(10))) / sqrt(3). - Vaclav Kotesovec, Sep 22 2020
Sum_{n>=1} 1/a(n) = 27/8. - Amiram Eldar, Sep 23 2020

A337801 Numbers of the form (2^i)*(5^j) or (3^i)*(5^j).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 15, 16, 20, 25, 27, 32, 40, 45, 50, 64, 75, 80, 81, 100, 125, 128, 135, 160, 200, 225, 243, 250, 256, 320, 375, 400, 405, 500, 512, 625, 640, 675, 729, 800, 1000, 1024, 1125, 1215, 1250, 1280, 1600, 1875, 2000, 2025, 2048, 2187, 2500
Offset: 1

Views

Author

Vaclav Kotesovec, Sep 22 2020

Keywords

Comments

Union of A003592 and A003593.

Links

Crossrefs

Cf. A003592, A003593, A257997, A258023, A337800.

Programs

  • Mathematica
    n = 10^4; Join[Table[2^i*5^j, {i, 0, Log[2, n]}, {j, 0, Log[5, n/2^i]}], Table[3^i*5^j, {i, 0, Log[3, n]}, {j, 0, Log[5, n/3^i]}]] // Flatten // Union (* Amiram Eldar, Sep 23 2020 *)

Formula

a(n) ~ exp(sqrt(2*log(2)*log(3)*log(5)*n / log(6))) / sqrt(5).
Sum_{n>=1} 1/a(n) = 25/8. - Amiram Eldar, Sep 23 2020
Showing 1-3 of 3 results.

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