A064459 -id:A064459 - cp's OEIS frontend

A053832 Sum of digits of n written in base 12.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 7, 8

Offset: 0

Henry Bottomley, Mar 28 2000

Also the fixed point of the morphism 0->{0,1,2,3,4,5,6,7,8,9,10,11}, 1->{1,2,3,4,5,6,7,8,9,10,11,12}, 2->{2,3,4,5,6,7,8,9,10,11,12,13}, etc. - Robert G. Wilson v, Jul 27 2006

			a(20) = 1 + 8 = 9 because 20 is written as 18 base 12.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Jeffrey O. Shallit, Problem 6450, Advanced Problems, The American Mathematical Monthly, Vol. 91, No. 1 (1984), pp. 59-60; Two series, solution to Problem 6450, ibid., Vol. 92, No. 7 (1985), pp. 513-514.
Robert Walker, Self Similar Sloth Canon Number Sequences.
Eric Weisstein's World of Mathematics, Duodecimal.
Eric Weisstein's World of Mathematics, Digit Sum.

Cf. A000120, A007953, A064459, A138530.

a053832 n = q 0 $ divMod n 12 where
   q r (0, d) = r + d
   q r (m, d) = q (r + d) $ divMod m 12
-- Reinhard Zumkeller, May 15 2011

Table[Plus @@ IntegerDigits[n, 12], {n, 0, 85}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 11}]] &, {0}, 2] (* Robert G. Wilson v, Jul 27 2006 *)

a(n)=if(n<1,0,if(n%12,a(n-1)+1,a(n/12)))

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(12n+i) = a(n)+i for 0 <= i <= 11.
a(n) = n-11*(Sum_{k>0} floor(n/12^k)) = n-11*A064459(n). (End)
a(n) = A138530(n,12) for n > 11. - Reinhard Zumkeller, Mar 26 2008
Sum_{n>=1} a(n)/(n*(n+1)) = 12*log(12)/11 (Shallit, 1984). - Amiram Eldar, Jun 03 2021

A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 4, 2, 1, 1, 1, 4, 2, 1, 1, 1, 1, 7, 2, 2, 1, 1, 1, 1, 7, 4, 2, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 5, 3, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 11, 6, 3, 3, 2, 2, 1, 1, 1

Offset: 2

Henry Bottomley, Dec 06 2003

			Rows start:
  1;
  1,1;
  3,1,1;
  3,1,1,1;
  4,2,1,1,1;
  4,2,1,1,1,1;
  7,2,2,1,1,1,1;
  7,4,2,1,1,1,1,1;
  8,4,2,2,1,1,1,1,1;
  ...

Zhuorui He, Table of n, a(n) for n = 2..11326
Wenguang Zhai, On the prime power factorization of n!, Journal of Number Theory, Volume 129, Issue 8, August 2009, Pages 1820-1836.

Columns include A011371, A054861, A054893, A027868, A054895, A054896, A054897, A054898, A054899, A064458, A064459, A090620, A054900.
Row sums: A078632.
Cf. A240236.

A090623[n_, k_] := Quotient[n - DigitSum[n, k], k - 1];
Table[A090623[n, k], {n, 2, 15}, {k, 2, n}] (* Paolo Xausa, Sep 02 2025 *)

T(n,k) = {my(s = 0, j = 1); while(p=n\k^j, s += p; j++); s;} \\ Michel Marcus, Feb 02 2016

T(n,k) = (n - sumdigits(n,k))/(k-1) \\ Zhuorui He, Aug 25 2025

For p prime, T(n, p) = A090622(n, p) is the number of times that p is a factor of n!.

T(n,k) = (n - A240236(n, k))/(k - 1). - Zhuorui He, Aug 25 2025

a(41) onward corrected by Zhuorui He, Aug 25 2025

A342696 a(n) = floor(n/12).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 8

Offset: 0

Wesley Ivan Hurt, May 18 2021

Agrees with A064459 for n < 144, but a(144) = 12 whereas A064459(144) = 13.

Index entries for linear recurrences with constant coefficients, signature (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Cf. A064459, A221912 (partial sums), A344420 (floor n/11).

Mathematica
```
Floor[Range[0, 200]/12]
```

G.f.: x^12 / ( (1+x)*(1+x^2)*(x^4-x^2+1)*(x^2-x+1)*(1+x+x^2)*(x-1)^2 ). - R. J. Mathar, Jul 08 2021

a(n) = a(n-1) + a(n-12) - a(n-13). - Wesley Ivan Hurt, Oct 29 2022

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A053832 Sum of digits of n written in base 12.

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A090623 Triangle of T(n,k) = [n/k] + [n/k^2] + [n/k^3] + [n/k^4] + ... for n, k > 1.

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A342696 a(n) = floor(n/12).

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