A362842 a(1) = 1; a(2) = 2; for n > 2, a(n) is the smallest positive number that has not yet appeared that shares a factor with a(n-1) when both a(n-1) and a(n) are read as numbers in bases from one more than the maximum digit in a(n-1) and a(n), up to base 10.

1, 2, 4, 6, 3, 9, 12, 24, 8, 20, 10, 30, 33, 11, 22, 26, 13, 39, 15, 48, 28, 14, 49, 7, 70, 16, 38, 19, 57, 69, 18, 56, 76, 36, 60, 40, 42, 21, 63, 66, 44, 46, 23, 92, 32, 64, 62, 31, 93, 27, 90, 5, 50, 55, 77, 84, 35, 80, 68, 17, 119, 34, 94, 47, 329, 91, 52, 96, 45, 95, 25, 190, 54, 98, 58, 29

Offset

1,2

Comments

This is a base variation of the EKG sequence A064413. Despite numbers with larger digits having to share a factor with a(n-1) in fewer bases than those with only small digits, and would therefore seemingly appear more frequently, the frequency of the digits 8 and 9, for example, in the first 200000 terms is the same as the smaller digits 0 to 7, so surprisingly this does not appear to influence the determination of a(n).

In the first 200000 terms the smallest unused number is 25411, which implies all numbers will eventually appear. In the same range the fixed points are 1, 2, 424, 507, 1261, 1577, 2461, 4311; it is likely no more appear.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 100000 terms. The green line is a(n) = n.

Example

a(7) = 12 as the maximum digit in a(6) = 9 and 12 is 9, so a(6) and a(7) are only read as base 10 numbers, and 12 is the smallest unused number which shares a factor with 9 in base 10.
a(8) = 24 as the maximum digit in a(7) = 12 and 24 is 4, and 12_k shares a factor with 24_k when they are read as numbers in all bases k = 5,6,7,8,9,10. No unused smaller number has this property, e.g. a(8) cannot equal 8 as a(7) in base 9 is 12_9 = 11, which does not share a factor with 8_9 = 8. This is the first term to differ from A064413.
a(9) = 8 as the maximum digit in a(8) = 24 and 8 is 8, and 24_k shares a factor with 8_k when they are read as numbers in all bases k = 9,10.

Crossrefs

Cf. A064413, A004053, A354087, A352763, A348086, A337687.

Sequence in context: A356851 A291577 A354434 * A338059 A258105 A331525

Adjacent sequences: A362839 A362840 A362841 * A362843 A362844 A362845

Keyword

nonn,base

Author

Scott R. Shannon, May 05 2023

Status

approved