A358533 Define a family of integer sequences S_0, S_1, S_2, ..., where S_0 = A000040 is the sequence of prime numbers and, for each k > 0, S_k is the result of making a "smoothing" pass through all the terms of S_(k-1) as follows: for every term other than the first, in ascending order, change its value by the minimum amount so that it will not differ from the mean of its two immediate neighbors by more than 1/2. {a(n)} is the limiting sequence S_oo.

2, 3, 5, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 44, 48, 53, 57, 62, 66, 70, 74, 79, 84, 90, 95, 99, 103, 107, 112, 118, 124, 130, 136, 142, 147, 152, 157, 162, 167, 172, 178, 184, 189, 194, 200, 206, 213, 219, 224, 229, 234, 239, 245, 251, 257, 262, 267, 272

Offset

1,1

Links

Table of n, a(n) for n=1..58.

Example

In the first table below, asterisks mark the terms of the sequence that change during a pass.
During the first pass, the 4th element moves from 7 to 8 (midway between its neighbors 5 and 11); 11 does not move, because (at this point in the pass) its neighbors are 8 and 13, whose mean is 10.5; 13 moves to 14, 19 to 20, 23 to 24, 29 to 28, 31 to 32, 41 to 40, 47 to 48, etc.
During the second pass, the 12th element moves from 37 moves to 36; 43 moves to 44, etc.
(During the third and subsequent passes, nothing below 106 moves.)
.
S_0  = { 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, ... }
                  *       *       *   *   *   *       *       *
S_1  = { 2, 3, 5, 8, 11, 14, 17, 20, 24, 28, 32, 37, 40, 43, 48, 53, ... }
                                                  *       *
S_2  = { 2, 3, 5, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 44, 48, 53, ... }
...
S_oo = {a(n)}
     = { 2, 3, 5, 8, 11, 14, 17, 20, 24, 28, 32, 36, 40, 44, 48, 53, ... }
.
The ASCII graphic below shows how the terms become more evenly spaced as a result of the first two passes.
.
            1         2         3         4         5         6
    234567890123456789012345678901234567890123456789012345678901234567
0:  ** * *   * *   * *   *     * *     *   * *   *     *     * *     *
1:  ** *  *  *  *  *  *   *   *   *    *  *  *    *    *   *    *    *
2:  ** *  *  *  *  *  *   *   *   *   *   *   *   *    *   *    *   *
...
oo: ** *  *  *  *  *  *   *   *   *   *   *   *   *    *   *    *   *

Crossrefs

Cf. A000040.

Sequence in context: A071894 A301892 A271876 * A078444 A332071 A225087

Adjacent sequences: A358530 A358531 A358532 * A358534 A358535 A358536

Keyword

nonn

Author

Jon E. Schoenfield, Nov 20 2022

Status

approved