A054898 -id:A054898 - cp's OEIS frontend

A053830 Sum of digits of (n written in base 9).

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

Offset: 0

Henry Bottomley, Mar 28 2000

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

			a(20) = 2+2 = 4 because 20 is written as 22 base 9.
From _Omar E. Pol_, Feb 23 2010: (Start)
It appears that this can be written as a triangle (see the conjecture in the entry A000120):
0;
1,2,3,4,5,6,7,8;
1,2,3,4,5,6,7,8,9,2,3,4,5,6,7,8,9,10,3,4,5,6,7,8,9,10,11,4,5,6,7,8,9,10,11,...
where the rows converge to A173529. (End)

Indranil Ghosh, Table of n, a(n) for n = 0..59049
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, Digit Sum.

Cf. A000120, A007953, A053735, A053737, A053824, A053828, A231684, A231685, A231686, A231687.

Cf. A173529. - Omar E. Pol, Feb 23 2010

[&+Intseq(n, 9):n in [0..100]]; // Marius A. Burtea, Aug 24 2019

Table[Plus @@ IntegerDigits[n, 9], {n, 0, 100}] (* or *)
Nest[ Flatten[ #1 /. a_Integer -> Table[a + i, {i, 0, 8}]] &, {0}, 3] (* Robert G. Wilson v, Jul 27 2006 *)

PARI
```
a(n)=if(n<1,0,if(n%9,a(n-1)+1,a(n/9)))
```

From Benoit Cloitre, Dec 19 2002: (Start)
a(0) = 0, a(9n+i) = a(n) + i for 0 <= i <= 8;
a(n) = n - 8*Sum_{k>=1} floor(n/9^k) = n - 8*A054898(n). (End)
a(n) = A138530(n,9) for n > 8. - Reinhard Zumkeller, Mar 26 2008
a(n) = Sum_{k>=0} A031087(n,k). - Philippe Deléham, Oct 21 2011
a(0) = 0; a(n) = a(n - 9^floor(log_9(n))) + 1. - Ilya Gutkovskiy, Aug 24 2019
Sum_{n>=1} a(n)/(n*(n+1)) = 9*log(9)/8 (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

A262927 a(n+9) = a(n) + 10*(n+4) + 9. a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=10, a(5)=15, a(6)=23, a(7)=30, a(8)=39.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 23, 30, 39, 49, 60, 72, 85, 99, 114, 132, 149, 168, 188, 209, 231, 254, 278, 303, 331, 358, 387, 417, 448, 480, 513, 547, 582, 620, 657, 696, 736, 777, 819, 862, 906, 951, 999, 1046, 1095, 1145, 1196, 1248, 1301, 1355, 1410, 1468, 1525

Offset: 0

Paul Curtz, Oct 04 2015

The main (or principal) sequence for the 11 steps recurrence is 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 30, 33, 36, ..., the partial sums of A054898.

a(n) mod 9 is a sequence of period 90.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (2,-1,0,0,0,0,0,0,1,-2,1).

Cf. A054898, A261327, A262397.

I:=[0,1,3,6,10,15,23,30,39]; [n le 9 select I[n] else (Self(n-9)+10*(n-6)+9): n in [1..60]]; // Vincenzo Librandi, Oct 06 2015

LinearRecurrence[{2, -1, 0, 0, 0, 0, 0, 0, 1, -2, 1}, {0, 1, 3, 6, 10, 15, 23, 30, 39, 49, 60}, 60] (* Vincenzo Librandi, Oct 06 2015 *)
RecurrenceTable[{a[n+9] == a[n] + 10*(n+4) + 9, a[0]=0, a[1]=1, a[2]=3, a[3]=6, a[4]=10, a[5]=15, a[6]=23, a[7]=30, a[8]=39},a,{n,0,1000}] (* G. C. Greubel, Oct 16 2015 *)

a(n) = numerator(((2*n)^2+4)/4)\9 + numerator(((2*n+1)^2+4)/4)\9;
vector(100, n, a(n-1)) \\ Altug Alkan, Oct 04 2015

concat(0, Vec(-x*(x^8+2*x^7-x^6+3*x^5+x^4+x^3+x^2+x+1)/((x-1)^3*(x^2+x+1)*(x^6+x^3+1)) + O(x^100))) \\ Colin Barker, Oct 04 2015

a(n)=((2*n+1)^2+4)\9+(n^2+1)\9 \\ Charles R Greathouse IV, Oct 16 2015

a(n) = A262397(2n) + A262397(2n+1).
a(n) = 2*a(n-1) - a(n-2) + a(n-9) - 2*a(n-10) + a(n-11), n>10.
G.f.: -x*(x^8+2*x^7-x^6+3*x^5+x^4+x^3+x^2+x+1) / ((x-1)^3*(x^2+x+1)*(x^6+x^3+1)). - Colin Barker, Oct 04 2015
a(n) = (5n^2 + 4n)/9 + O(1), or more precisely (5n^2 + 4n + 3)/9 <= a(n) <= (5n^2 + 4n - 10)/9. - Charles R Greathouse IV, Oct 16 2015

A381886 Triangle read by rows: T(n, k) = Sum_{j=1..floor(log[k](n))} floor(n / k^j) if k >= 2, T(n, 1) = n, T(n, 0) = 0^n.

Original entry on oeis.org

1, 0, 1, 0, 2, 1, 0, 3, 1, 1, 0, 4, 3, 1, 1, 0, 5, 3, 1, 1, 1, 0, 6, 4, 2, 1, 1, 1, 0, 7, 4, 2, 1, 1, 1, 1, 0, 8, 7, 2, 2, 1, 1, 1, 1, 0, 9, 7, 4, 2, 1, 1, 1, 1, 1, 0, 10, 8, 4, 2, 2, 1, 1, 1, 1, 1, 0, 11, 8, 4, 2, 2, 1, 1, 1, 1, 1, 1, 0, 12, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 0

Peter Luschny, Apr 03 2025

			Triangle starts:
  [ 0] 1;
  [ 1] 0,  1;
  [ 2] 0,  2,  1;
  [ 3] 0,  3,  1, 1;
  [ 4] 0,  4,  3, 1, 1;
  [ 5] 0,  5,  3, 1, 1, 1;
  [ 6] 0,  6,  4, 2, 1, 1, 1;
  [ 7] 0,  7,  4, 2, 1, 1, 1, 1;
  [ 8] 0,  8,  7, 2, 2, 1, 1, 1, 1;
  [ 9] 0,  9,  7, 4, 2, 1, 1, 1, 1, 1;
  [10] 0, 10,  8, 4, 2, 2, 1, 1, 1, 1, 1;
  [11] 0, 11,  8, 4, 2, 2, 1, 1, 1, 1, 1, 1;
  [12] 0, 12, 10, 5, 3, 2, 2, 1, 1, 1, 1, 1, 1;

Jeffrey C. Lagarias and Wijit Yangjit, The factorial function and generalizations, extended, arXiv:2310.12949 [math.NT], 2023.
A. M. Oller-Marcen and J. Maria Grau, On the Base-b Expansion of the Number of Trailing Zeros of b^k!, J. Int. Seq. 14 (2011) 11.6.8.

Cf. A011371 (column 2), A054861 (column 3), A054893 (column 4), A027868 (column 5), A054895 (column 6), A054896 (column 7), A054897 (column 8), A054898 (column 9), A078651 (row sums).

Cf. A078632, A078567, A153216, A366471.

T := (n, b) -> local i; ifelse(b = 0, b^n, ifelse(b = 1, n, add(iquo(n, b^i), i = 1..floor(log(n, b))))): seq(seq(T(n, b), b = 0..n), n = 0..12);
# Alternative:
T := (n, k) -> local j; ifelse(k = 0, k^n, ifelse(k = 1, n, add(padic:-ordp(j, k), j = 1..n))): for n from 0 to 12 do seq(T(n, k), k = 0..n) od;

T[n_, 0] := If[n == 0, 1, 0]; T[n_, 1] := n;
T[n_, k_] := Last@Accumulate[IntegerExponent[Range[n], k]];
Table[T[n, k], {n, 0, 12}, {k, 0, n}] // MatrixForm
(* Alternative: *)
T[n_, k_] := Sum[Floor[n/k^j], {j, Floor[Log[k, n]]}]; T[n_, 1] := n; T[n_, 0] := 0^n; T[0, 0] = 1; Flatten@ Table[T[n, k], {n, 0, 12}, {k, 0, n}] (* Michael De Vlieger, Apr 03 2025 *)

T(n,k) = if (n==0, 1, if (n==1, k, if (k==0, 0, if (k==1, n, sum(j=1, n, valuation(j, k))))));
row(n) = vector(n+1, k, T(n,k-1)); \\ Michel Marcus, Apr 04 2025

from math import log
def T(n: int, b: int) -> int:
    return (b**n if b == 0 else n if b == 1 else
        sum(n // (b**i) for i in range(1, 1 + int(log(n, b)))))
print([[T(n, b) for b in range(n+1)] for n in range(12)])

def T(n, b): return (b^n if b == 0 else n if b == 1 else sum(valuation(j, b) for j in (1..n)))
print(flatten([[T(n, b) for b in range(n+1)] for n in srange(13)]))

T(n, k) = Sum_{j=1..n} valuation(j, k) for n >= 2.

cp's OEIS Frontend

A053830 Sum of digits of (n written in base 9).

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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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A262927 a(n+9) = a(n) + 10*(n+4) + 9. a(0)=0, a(1)=1, a(2)=3, a(3)=6, a(4)=10, a(5)=15, a(6)=23, a(7)=30, a(8)=39.

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A381886 Triangle read by rows: T(n, k) = Sum_{j=1..floor(log[k](n))} floor(n / k^j) if k >= 2, T(n, 1) = n, T(n, 0) = 0^n.

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