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.

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

0,5

Links

Table of n, a(n) for n=0..90.

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.

Formula

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

Example

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;

Maple

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;

Mathematica

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 *)

Prog

(Python)
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)])
(SageMath)
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)]))
(PARI) 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

Crossrefs

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.

Sequence in context: A376498 A176808 A327029 * A167192 A335435 A363048

Adjacent sequences: A381883 A381884 A381885 * A381887 A381888 A381889

Keyword

nonn,tabl

Author

Peter Luschny, Apr 03 2025

Status

approved