A353121 Jackpot integers: number of steps needed by n to win a "jackpot" (see the explanation below in the Comments and Example sections) or -1 if this will never occur.

-1, 6, 2, 3, 1, 2, 1, 10, 2, 6, -1, 2, 6, 17, 3, 2, 3, 5, 2, 4, -1, 3, 17, 16, 2, 17, 2, 2, 1, 3, -1, 1, 3, 2, 4, 3, 1, 2, 4, 8, -1, 2, 2, 17, 3, 6, 3, 5, 2, 4, -1, 1, 3, 2, 1, 3, 11, 2, 4, 8, -1, 10, 5, 2, 2, 5, 2, 3, 3, 1, -1, 2, 2, 1, 4, 2, 4, 3, 4, 3, -1, 6, 4, 3, 8, 4, 8, 1, 3, 1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, 5, 8, 19, 34, 8, 34

Offset

10,2

Comments

Write vertically a number (like 419) and try to win the jackpot (using the hereunder martingale):

.

(4) - a - d - g - j - m

(1) - b - e - h - k - ...

(9) - c - f - i - l - ...

.

i) compute the terms a, b, c, d, e, ... in the order shown above;

ii) the sequence a, b, c, d, e, ... is always extended with the smallest prime not occurring before, such that ...

iii) ... in each row, the first (parenthesized) digit of the next term is the rightmost digit of the preceding term;

iv) a jackpot is won when all terms in a column have the same rightmost digit.

.

4 - (4)1 - (1)3 - (3) - (3)1 - (1)01 - (1)03 - (3)07 - (7)09

1 - (1)1 - (1)7 - (7)1 - (1)9 - (9)07 - (7)01 - (1)07 - (7)19

9 - (9)7 - (7) - (7)3 - (3)7 - (7)9 - (9)11 - (1)09 - (9)19 = jackpot!

.

Indeed, we have the same digit (9) ending all the terms of the last column.

And as it takes 8 steps to win the jackpot with 419, we assign 8 to 419.

Links

Table of n, a(n) for n=10..116.

Eric Angelini, Jackpot numbers, Jackpot numbers, personal blog.

Example

10 => -1 (no number that contains a digit 0 can win a jackpot; we assign -1 to 10 to mark this impossibility - see the two rows generated by 10 hereunder):
   1 - 11
   0 - ? (no prime starts with 0)
11 => 6 steps
   1 - 11 - 17 - 7  - 71 - 101 - 103
   1 - 13 - 3  - 31 - 19 - 97  - 73
12 => 2 steps
   1 - 11 - 13
   2 - 2  - 23
13 => 3 steps
   1 - 11 - 13 - 37
   3 - 3  - 31 - 17
14 => 1 step
   1 - 11
   4 - 41
15 => 2 steps
   1 - 11 - 13
   5 - 5  - 53
16 => 1 step
   1 - 11
   6 - 61
17 => 10 steps
   1 - 11 - 13 - 3  - 31 - 19 - 97 - 701 - 101 - 103 - 307
   7 - 7  - 71 - 17 - 73 - 37 - 79 - 907 - 709 - 911 - 107
Etc. The number of successive steps form the sequence.

Crossrefs

Cf. A000040.

Sequence in context: A247672 A188726 A272354 * A196552 A062614 A155527

Adjacent sequences: A353118 A353119 A353120 * A353122 A353123 A353124

Keyword

sign,base

Author

Eric Angelini and Carole Dubois, Apr 24 2022

Status

approved