A013584 Smallest m such that 0!+1!+...+(m-1)! is divisible by n, or 0 if no such m exists.

1, 2, 0, 3, 4, 0, 6, 0, 0, 4, 6, 0, 0, 6, 0, 0, 5, 0, 7, 0, 0, 6, 7, 0, 0, 0, 0, 0, 0, 0, 12, 0, 0, 5, 0, 0, 22, 7, 0, 0, 16, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 55, 12, 0, 0, 0, 0, 0, 0, 0, 0, 54, 0, 42, 22, 0, 0, 6, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Author

Michael R. Mudge (Amsorg(AT)aol.com)

Keywords

• nonn

Comments

From Robert Israel, Nov 14 2016: (Start)

a(n) < n for n > 2.

If a(n) = 0, then a(mn) = 0 for all m>=2. (End)

References

• M. R. Mudge, Smarandache Notions Journal, University of Craiova, Vol. VII, No. 1, 1996.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A003422, A013585, A049044, A049045, A275608.

Programs

Maple

f:= proc(n) local t,r,m;
  r:= 1; t:= 1;
  for m from 1 do
    r:= r*m mod n;
    if r = 0 then return 0 fi;
    t:= t + r mod n;
    if t = 0 then return m+1 fi;
  od;
end proc:
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Nov 14 2016

Mathematica

a[n_] := Module[{t, r, m}, r = 1; t = 1; For[m = 1, True, m++, r = Mod[r*m, n]; If[r == 0, Return[0]]; t = Mod[t+r, n]; If[t == 0, Return[m+1]]]];
Array[a, 100] (* Jean-François Alcover, Apr 12 2019, after Robert Israel *)