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.
a(1) = floor(e^(1!)) = floor(e) = 2.
Table[Floor[E^n!], {n, 1, 7}]
default(realprecision, 100); vector(5, n, floor(exp(n!))) \\ Michel Marcus, Aug 06 2015
a(3) = 8 since floor(e^3) = 20, floor(Sum_{k=0..8} (n^k)/(k!)) = 20 and "8" is the minimum because floor(Sum_{k=0..7} (n^k)/(k!)) = 19.
f[n_, m_] := Floor[Sum[(n^k)/(k!), {k, 0, m}]] - Floor[E^n];
a[n_] := Min[Flatten[Position[Table[f[n, m], {m, 0, 150}], 0]]] - 1;
Table[a[n], {n, 1, 50}]
a(n) = my(m=0, x=floor(exp(n)), y=1); while(floor(y) != x, m++; y += n^m/m!); m; \\ Michel Marcus, Apr 14 2023
e^10 = 22026.46579480671... 22026 in binary is 101011000001010 so a(10)=15.
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