A389674 If n is a power of 10, then leave n untouched, else rewrite n in the base given by the sum of its digits, and 'read' those new digits in base 10.

1, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1011, 110, 31, 24, 23, 22, 21, 20, 19, 10100, 210, 112, 43, 40, 34, 32, 30, 28, 27, 1010, 133, 112, 53, 46, 43, 40, 37, 35, 33, 220, 131, 110, 61, 54, 50, 46, 43, 40, 40, 200, 123, 103, 65, 60, 55, 51, 49, 46, 43, 140, 115

Offset

1,2

Comments

The "rebase-and-read" operation is as follows:

1: Take n written in base 10, n = Sum_{i>=0} d_i * 10^i, and sum its digits together to obtain b = Sum_{i>=0} d_i.

2: If b = 1, then a(n) = n. Otherwise, "rebase": take n in base b so n = Sum_{i>=0} c_i * b^i and then "read" the digits c_i as base 10 to obtain a(n) = Sum_{i>=0} c_i * 10^i.

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Example

For n = 8, its sum of digits is 8 and 8 = 1 * 8^1 + 0 * 8^0, so a(8) = 10.
For n = 100, its sum of the digits is 1, so by definition a(100) = 100.
For n = 1506, its sum of digits is 12 and 1506 = 10 * 12^2 + 5 * 12^1 + 6 * 12^0, so a(1506) = 10 * 10^2 + 5 * 10^1 + 6 * 10^0 = 1056.

Mathematica

a[n_]:=If[IntegerQ[Log10[n]], n, FromDigits[IntegerDigits[n, DigitSum[n]]]]; Array[a, 61] (* James C. McMahon, Oct 16 2025 *)

Prog

(Python)
from sympy.ntheory import digits
def a(n): return n if (b:=sum(d:=list(map(int, str(n)))))<2 else sum(di*10**i for i, di in enumerate(digits(n, b)[:0:-1]))
print([a(n) for n in range(1, 62)]) # Michael S. Branicky, Oct 10 2025
(PARI) a(n) = my(s=sumdigits(n)); if (s==1, n, fromdigits(digits(n, s))); \\ Michel Marcus, Oct 13 2025

Crossrefs

Cf. A007953, A011557, A326833 (fixed points).

Sequence in context: A245403 A091837 A316650 * A216875 A166710 A211872

Adjacent sequences: A389671 A389672 A389673 * A389675 A389676 A389677

Keyword

base,easy,nonn

Author

Daniel Baldwin, Oct 10 2025

Status

approved