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A204932 Least k such that n divides k!-j! for some j satisfying 1<=j

%I A204932 #10 Nov 11 2024 01:59:20
%S A204932 2,3,4,3,3,4,5,5,4,6,4,5,9,6,6,5,5,4,5,6,6,4,4,5,6,9,7,8,6,6,10,5,7,6,
%T A204932 8,7,13,5,9,6,7,6,11,7,7,8,10,5,8,6,6,9,12,7,8,8,5,6,5,6,16,10,8,9,14,
%U A204932 7,8,11,8,8,9,7,8,13,6,7,12,9,14,7,10,7,19,8,11,11,6,7,10,7
%N A204932 Least k such that n divides k!-j! for some j satisfying 1<=j<k.
%C A204932 See A204892 for a discussion and guide to related sequences.
%H A204932 Robert Israel, <a href="/A204932/b204932.txt">Table of n, a(n) for n = 1..10000</a>
%e A204932 1 divides 2!-1!, so a(1)=2
%e A204932 2 divides 3!-2!, so a(2)=3
%e A204932 3 divides 4!-3!, so a(3)=4
%e A204932 13 divides 9!-4!, so a(13)=9
%p A204932 f:= proc(n) local t, k, V;
%p A204932   t:= 1:
%p A204932   for k from 1 do
%p A204932     t:= t*k mod n;
%p A204932     if assigned(V[t]) then return k else V[t]:= k fi
%p A204932   od
%p A204932 end proc:
%p A204932 map(f, [$1..100]); # _Robert Israel_, Nov 11 2024
%t A204932 s[n_] := s[n] = n!; z1 = 80; z2 = 60;
%t A204932 Table[s[n], {n, 1, 30}]     (* A000142 *)
%t A204932 u[m_] := u[m] = Flatten[Table[s[k] - s[j], {k, 2, z1}, {j, 1, k - 1}]][[m]]
%t A204932 Table[u[m], {m, 1, z1}]     (* A204930 *)
%t A204932 v[n_, h_] := v[n, h] = If[IntegerQ[u[h]/n], h, 0]
%t A204932 w[n_] := w[n] = Table[v[n, h], {h, 1, z1}]
%t A204932 d[n_] := d[n] = First[Delete[w[n], Position[w[n], 0]]]
%t A204932 Table[d[n], {n, 1, z2}]     (* A204931 *)
%t A204932 k[n_] := k[n] = Floor[(3 + Sqrt[8 d[n] - 1])/2]
%t A204932 m[n_] := m[n] = Floor[(-1 + Sqrt[8 n - 7])/2]
%t A204932 j[n_] := j[n] = d[n] - m[d[n]] (m[d[n]] + 1)/2
%t A204932 Table[k[n], {n, 1, z2}]        (* A204932 *)
%t A204932 Table[j[n], {n, 1, z2}]        (* A204933 *)
%t A204932 Table[s[k[n]], {n, 1, z2}]     (* A204934 *)
%t A204932 Table[s[j[n]], {n, 1, z2}]     (* A204935 *)
%t A204932 Table[s[k[n]] - s[j[n]], {n, 1, z2}]     (* A204936 *)
%t A204932 Table[(s[k[n]] - s[j[n]])/n, {n, 1, z2}] (* A204937 *)
%Y A204932 Cf. A000142, A204892.
%K A204932 nonn
%O A204932 1,1
%A A204932 _Clark Kimberling_, Jan 21 2012