A344853 -id:A344853 - cp's OEIS frontend

A356384 For any n >= 0, let x_n(1) = n, and for any b > 1, x_n(b) = x_n(b-1) minus the sum of digits of x_n(b-1) in base b; a(n) is the least b such that x_n(b) = 0.

1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 12, 12, 12, 12, 13, 13, 13, 13, 13

Offset: 0

Rémy Sigrist, Aug 05 2022

This sequence is well defined: for any n >= 0: if x_n(b) > 0, then x_n(b+1) < x_n(b), and we must eventually reach 0.
This sequence is weakly increasing; this is related to the fact that for any base b > 1, k -> (k minus the sum of digits of k in base b) is weakly increasing.
Note that some values (like 7) do not appear in this sequence (see also A356386).

			For n = 42:
- we have:
      b  x(b)
      -  ----
      1    42
      2    39
      3    36
      4    33
      5    28
      6    20
      7    12
      8     7
      9     0
- so a(42) = 9.

Rémy Sigrist, Table of n, a(n) for n = 0..10000
Rémy Sigrist, Colored scatterplot of (n, x_n(b)) for n <= 1000 and b = 1..a(n) (the color is function of b)
Rémy Sigrist, PARI program

Cf. A011371, A066568, A071542, A261231, A344853, A356386.

PARI
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See Links section.
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A346502 a(n) = 3n - (sum of digits of 3n in base 3).

Original entry on oeis.org

0, 2, 4, 8, 10, 12, 16, 18, 20, 26, 28, 30, 34, 36, 38, 42, 44, 46, 52, 54, 56, 60, 62, 64, 68, 70, 72, 80, 82, 84, 88, 90, 92, 96, 98, 100, 106, 108, 110, 114, 116, 118, 122, 124, 126, 132, 134, 136, 140, 142, 144, 148, 150, 152, 160, 162, 164, 168, 170, 172

Offset: 0

Bernard Schott, Jul 21 2021

Terms of A344853 without repetition.
All terms are even.
A new largest gap between 2 consecutive terms is obtained between a(3^m-1) and a(3^m), m >= 0 (see formula).
In base 2, A005187(n) = 2n - (sum of digits of 2n in base 2) is also the exponent of the largest power of 2 dividing (2n)!, but here the exponent of the largest power of 3 dividing (3n)! is not a(n) but A004128(n).

			a(8) = 24 - (sum of digits of 24 in base 3); 24_10 = 220_3 and 2+2+0 = 4, so a(8) = 24-4 = 20.

Cf. A005187 (similar, with base 2).

Cf. A004128, A053735, A344853.

a[n_] := 3*n - Plus @@ IntegerDigits[3*n, 3]; Array[a, 100, 0] (* Amiram Eldar, Jul 22 2021 *)

a(n) = 3*n - sumdigits(n,3); \\ Kevin Ryde, Jul 21 2021

from sympy.ntheory.digits import digits
def a(n): return 3*n - sum(digits(3*n, 3)[1:])
print([a(n) for n in range(60)]) # Michael S. Branicky, Jul 28 2021

a(n) = 3*n - A053735(3*n).
a(n) = 2*A004128(n).
a(n) = A344853(3n).
a(3^n) - a(3^n-1) = 2*(n+1).

cp's OEIS Frontend

A356384 For any n >= 0, let x_n(1) = n, and for any b > 1, x_n(b) = x_n(b-1) minus the sum of digits of x_n(b-1) in base b; a(n) is the least b such that x_n(b) = 0.

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