A363838
a(n) = Product_{b=2..n} b^gamma(n, b) where gamma(n, b) = Sum_{i>=1} floor(n/b^i).
Original entry on oeis.org
1, 1, 2, 6, 96, 480, 17280, 120960, 30965760, 2508226560, 250822656000, 2759049216000, 9535274090496000, 123958563176448000, 24295878382583808000, 5466572636081356800000, 179128652139113899622400000, 3045187086364936293580800000, 53278593263040925392489676800000
Offset: 0
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f(n, b) = sum(i=1, logint(n, b), n\b^i);
a(n) = prod(b=2, n, b^f(n,b));
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from math import prod
from sympy import integer_log
def A363838(n): return prod(b**sum(n//b**i for i in range(1,integer_log(n,b)[0]+1)) for b in range(2,n+1)) # Chai Wah Wu, Oct 20 2023
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
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;
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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;
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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 *)
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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
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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)])
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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)]))
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