A393112 Integers k >= 4 such that 2^k is record-close to a base-10 palindrome, in the sense that A393111(k)/2^(k/2) sets a new minimum.

4, 5, 9, 16, 28, 41, 81, 82, 107, 141, 1311, 1830, 2253, 2260, 4520, 5857, 9894, 69180, 77191, 268687

Offset

1,1

Comments

A393111(k) is typically close to 2^(k/2), so this sequence gives values of k for which the ratio of A393111(k) and the typical value sets new minima.

For large n, the numbers 2^a(n) have a significant palindromic part around the center of their decimal representations (see example table).

a(21) > 500000 (if it exists).

Links

Table of n, a(n) for n=1..20.

Example

 n | k=a(n) |          2^k          | closest palindrome p  | |2^k-p|/2^(k/2)
---+--------+-----------------------+-----------------------+----------------
 1 |      4 |          1 6          |          1 1          |  1.25
 2 |      5 |          3 2          |          3 3          |  0.176776...
 3 |      9 |          512          |          515          |  0.132582...
 4 |     16 |         65536         |         65556         |  0.078125
 5 |     28 |       268435456       |       268434862       |  0.036254...
 6 |     41 |     2199023255552     |     2199023209912     |  0.030777...
 7 |     81 | 2...1639229258349...2 | 2...1639229229361...2 |  0.018642...
 8 |     82 | 4...3278458516698...4 | 4...3278458548723...4 |  0.014562...
 9 |    107 | 1...2921336339157...8 | 1...2921336331292...1 |  0.006174...
10 |    141 | 2...7892691964784...2 | 2...7892691962987...2 |  0.001076...
11 |   1311 | 4...0168529260435...8 | 4...0168529258610...4 |  0.000863...
12 |   1830 | 7...5342877784695...4 | 7...5342877782435...7 |  0.000815...
13 |   2253 | 1...3715484845960...2 | 1...3715484845173...1 |  0.000611...
14 |   2260 | 2...5558206028299...6 | 2...5558206028555...2 |  0.000175...
15 |   4520 | 4...1910463640039...6 | 4...1910463640191...4 |  0.000071...
16 |   5857 | 1...508550 055793...2 | 1...508550 055805...1 |  0.000032...
17 |   9894 | 2...3072624262737...4 | 2...3072624262703...2 |  0.000021...
18 |  69180 | 1...425598 895528...6 | 1...425598 895524...1 |  0.000009...
19 |  77191 | 6...0234853584326...8 | 6...0234853584320...6 |  0.000002...
20 | 268687 | 7...2052578752505...8 | 7...2052578752502...7 |  0.000001...

Crossrefs

Cf. A393111.

Sequence in context: A051216 A255978 A069089 * A143096 A280225 A153058

Adjacent sequences: A393109 A393110 A393111 * A393113 A393114 A393115

Keyword

nonn,base,more

Author

Pontus von Brömssen, Feb 02 2026

Status

approved