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.

A381226 a(n) is the number of distinct positive integers that can be obtained by starting with n!, and optionally applying the operations square root, floor, and ceiling, in any order.

Original entry on oeis.org

1, 2, 4, 6, 7, 8, 8, 9, 10, 10, 10, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14, 14, 14, 14, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 17, 17, 17, 17, 17, 17, 17, 17, 17, 17, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18
Offset: 1

Views

Author

N. J. A. Sloane, Feb 24 2025

Keywords

Comments

This sequence, A381227, and A381228 arose in connection with the problem of showing that every positive integer can be represented using a single 4. Hans Havermann has pointed out that A139004 is related to this question and has many references. - N. J. A. Sloane, Feb 25 2025

Examples

			For n = 8, 8! = 40320; sqrt(40320) = 200.798..., floor and ceiling give 200 and 201. Sqrt(200) = 14.142..., and floor and ceiling give 14 and 15. From 14 we get 3 and 4; from 3 we get 1 and 2. 15 and 4 give nothing more. In all, we get a(8) = 9 different numbers: 40320, 200, 201, 14, 15, 3, 4, 1, 2.
Note that at each step, we must consider three "parents": if x was a term at the previous step, we get floor(sqrt(x)), sqrt(x), and ceiling(sqrt(x)) as potential parents at the next step.

Crossrefs

Cf. A139004, A381227-A381229.
Motivated by trying to understand A000319.

Programs

  • PARI
    f(n) = my(t); if(n<4, [1..n], t=sqrtint(n); if(issquare(n), concat(f(t), n), Set(concat([f(t), f(t+1), [n]]))));
a(n) = #f(n!); \\ Jinyuan Wang, Feb 25 2025

Extensions

More terms from Jinyuan Wang, Feb 25 2025

A381227 Irregular triangle read by rows: row n lists the A381226(n) numbers constructed in the definition of A381226, in increasing order.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 6, 1, 2, 3, 4, 5, 24, 1, 2, 3, 4, 10, 11, 120, 1, 2, 3, 5, 6, 26, 27, 720, 1, 2, 3, 8, 9, 70, 71, 5040, 1, 2, 3, 4, 14, 15, 200, 201, 40320, 1, 2, 3, 4, 5, 24, 25, 602, 603, 362880, 1, 2, 3, 6, 7, 43, 44, 1904, 1905, 3628800, 1, 2, 3, 8, 9, 79, 80, 6317, 6318, 39916800
Offset: 1

Views

Author

N. J. A. Sloane, Feb 25 2025

Keywords

Examples

			Triangle begins:
   1;
   1, 2;
   1, 2, 3, 6;
   1, 2, 3, 4,  5, 24;
   1, 2, 3, 4, 10, 11, 120;
   1, 2, 3, 5,  6, 26,  27,  720;
   1, 2, 3, 8,  9, 70,  71, 5040;
   1, 2, 3, 4, 14, 15, 200,  201, 40320;
   1, 2, 3, 4,  5, 24,  25,  602,   603,  362880;
   1, 2, 3, 6,  7, 43,  44, 1904,  1905, 3628800;
   ...

Crossrefs

Cf. A139004, A381226, A381228, A381229.

A381228 Smallest k such that n appears in row k of the triangle in A381227, or -1 if n never appears in A381227.

Original entry on oeis.org

1, 2, 3, 4, 4, 3, 10, 7, 7, 5, 5, 12, 12, 8, 8, 13, 13, 21, 36, 22, 22, 37, 14, 4, 9, 6, 6, 39, 39, 24, 24, 15, 15, 69, 41, 41, 25, 25, 42, 42, 72, 72, 10, 10, 43, 16, 16, 74, 128, 44, 44, 75, 130, 76, 76, 27, 27, 77, 77, 134, 134, 78, 46, 46, 17, 17, 79, 79, 28
Offset: 1

Views

Author

N. J. A. Sloane, Feb 25 2025

Keywords

Comments

Conjecture: Every positive integer appears in A381227.

Examples

			14 first appears in row 8 of A381227, so a(14) = 8.

Crossrefs

Cf. A139004, A381226, A381227, A381229.

Extensions

More terms from Jinyuan Wang, Feb 25 2025
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.