A380435 Erase digit 0 from decimal expansion of n. Then repeatedly apply the number of divisor function (A000005) onto each digit until a stationary value is reached. a(n) is the final stationary value (if it is reached for all digits).

1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 11, 12, 12, 12, 12, 12, 12, 12, 12, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22, 22, 22, 22, 2, 21, 22, 22, 22, 22, 22

Offset: 1

Author

Ctibor O. Zizka, Jan 24 2025

Keywords

• nonn
• base

Comments

The number of iterations is 0, 1, 2, 3 for numbers containing the highest digits (1, 2), (3,5,7), (4, 9), (6, 8). n >= a(n) >= 1.

Examples

			For n = 21 a(21) = 21.
For n = 408 we iterate 48 --> 34 --> 23 --> 22, thus, after 3 iterations, a(408) = 22.
		

Crossrefs

Cf. A002275, A002276 - A002283, A004719, A007931, A111066.

Programs

Mathematica

a[n_] := FromDigits[IntegerDigits[n] /. {0 -> Nothing, ?(# > 1 &) -> 2}]; Array[a, 100] (* _Amiram Eldar, Jan 24 2025 *)

Formula

a(A007931(n)) = A007931(n).

For r = 1, k >= 0:

a(10^k) = 1

a((10^k - 1)/9) = (10^k - 1)/9.

For r from [2, 9], k >= 0:

a(r*10^k) = 2.

a(r*(10^k - 1)/9) = 2*(10^k - 1)/9.