This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
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;
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)]))
Array begins: 1 1 1 3 1 1 3 1 1 1 4 2 2 1 2 ...
T[n_, k_] := IntegerExponent[n!, k];
Table[T[n, k], {n, 2, 15}, {k, 2, n}] // Flatten (* Jean-François Alcover, Sep 15 2021 *)
T(n,k) = valuation(n!, k); \\ Michel Marcus, Sep 15 2021
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