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.
Do[s=Mod[n!, 2^n]; If[IntegerQ[Log[2, s]], Print[Log[2, s]]], {n, 1, 1000}]
Table[Mod[n!,3^n],{n,0,30}] (* Harvey P. Dale, Apr 01 2018 *)
a(n) = n! % 3^n; \\ Michel Marcus, Jan 22 2021
import math print([math.factorial(n)%(3**n) for n in range(99)])
a(n) = n! % 4^n; \\ Michel Marcus, Jan 22 2021
import math print([math.factorial(n)%(4**n) for n in range(99)])
Not rarely,consecutive integers are in the sequence like {19,20,21}, providing residues {65536,262144,262144}.
t = {}; Do[s=Mod[n!, 2^n]; If[IntegerQ[Log[2, s]], AppendTo[t, n]], {n, 300}]; t
Select[Range[300],IntegerQ[Log2[Mod[#!,2^#]]]&] (* Harvey P. Dale, Aug 04 2021 *)
Do[s=Mod[n!, 2^n]; If[IntegerQ[Log[2, s]], Print[n-Log[2, s]]], {n, 1, 1000}]
n\k | 2 3 4 5 6 -----+--------------------------------------------- 2 | 2; 3 | 6; 4 | 8, 24; 5 | 24, 120; 6 | 16, 720; 7 | 48, 666, 5040; 8 | 128, 954, 40320; 9 | 384, 8586, 100736, 362880; 10 | 768, 26811, 483072, 3628800; 11 | 1280, 58725, 2168064, 39916800; 12 | 3072, 173259, 9239552, 234860975, 479001600;
row[n_] := Module[{k = 1, s = {}}, While[k^n <= n!, k++; AppendTo[s, Mod[n!, k^n]]]; s]; Table[row[n], {n, 2, 12}] // Flatten (* Amiram Eldar, Apr 28 2021 *)
def f(n)
return 1 if n < 2
(1..n).inject(:*)
end
def A(n)
m = f(n)
ary = []
(2..n).each{|i|
j = i ** n
ary << m % j
break if m <= j
}
ary
end
def A340810(n)
(2..n).map{|i| A(i)}.flatten
end
p A340810(12)