A013585 Smallest m such that 1!+...+m! is divisible by 2n+1, or 0 if no such m exists.

1, 2, 0, 0, 3, 4, 0, 0, 5, 0, 0, 12, 0, 7, 19, 0, 4, 0, 24, 0, 32, 19, 0, 0, 0, 5, 20, 0, 0, 0, 0, 0, 0, 20, 12, 0, 7, 0, 0, 57, 7, 0, 0, 19, 0, 0, 0, 0, 6, 8, 83, 0, 0, 15, 33, 24, 0, 0, 0, 0, 12, 32, 0, 38, 19, 9, 0, 0, 0, 23, 0, 0, 0, 0, 70, 71, 5, 0, 57, 20, 0, 17, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 0, 0, 0, 28

Offset: 0

Author

Michael R. Mudge (Amsorg(AT)aol.com), additional terms from Allan C. Wechsler

Keywords

• nonn

Comments

From Robert Israel, Nov 14 2016: (Start)

a(n) < 2*n for n > 1.

If a(n) = 0, then a((2*k+1)*n + k) = 0 for all k >= 0.

(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 = 0..10000

Crossrefs

Cf. A003422, A013584.

Programs

Maple

f:= proc(n) local t,r,m;
  r:= 1; t:= 0;
  for m from 1 do
    r:= r*m mod (2*n+1);
    if r = 0 then return 0 fi;
    t:= t + r mod (2*n+1);
    if t = 0 then return m fi;
  od;
end proc:
f(0):= 1:
map(f, [$0..100]); # Robert Israel, Nov 14 2016

Mathematica

a[n_] := Module[{t, r, m}, r = 1; t = 0; For[m = 1, True, m++, r = Mod[r m, 2 n + 1]; If[r == 0, Return[0]]; t = Mod[t + r, 2 n + 1]; If[t == 0, Return[m]]]];
a[0] = 1;
a /@ Range[0, 100] (* Jean-François Alcover, Jul 19 2020, after Maple *)