1,2

The factorials, where Smarandache(n!)/n! = 1/(n-1)!, appear to form a subsequence. The numbers A007672(a(n)) are small.

a(n) is either even, 19k, 23k or R*k, where R is a repunit prime. For example at 2.19.23=874, the corresponding repunit is divisible with 3 repunit primes. - Labos E. (labos(AT)ana.sote.hu), Jun 04 2004

A. J. Kempner, Concerning the smallest integer m! divisible by a given integer n, Amer. Math. Monthly 25 (1918) 204-210.

Eric Weisstein's World of Mathematics, Smarandache Function

The first 5 incrementally smallest ratios Smarandache(n)/n are 1, 1/2, 1/3, 1/4, 1/6. They occur at n = 1, 6, 12, 20, 24.

(Smarandache[n_] := (m=1; While[ !IntegerQ[m!/n], m++ ]; m); M = {}; L = {}; Do[With[{s = Smarandache[n]}, If[s/n < Min[M], M = Append[M, s/n]; L = Append[L, n]]], {n, 100}]; L)

Smarandache[1] := 1; Smarandache[n_] := Max[Smarandache @@@ FactorInteger[n]]; Smarandache[p_, 1] := p; Smarandache[p_, alpha_] := Smarandache[p, alpha] = Module[{a, k, r, i, nu, k0 = alpha(p - 1)}, i = nu = Floor[Log[p, 1 + k0]]; a[1] = 1; a[n_] := (p^n - 1)/(p - 1); k[nu] = Quotient[alpha, a[nu]]; r[nu] = alpha - k[nu]a[nu]; While[r[i] > 0, k[i - 1] = Quotient[r[i], a[i - 1]]; r[i - 1] = r[i] - k[i - 1]a[i - 1]; i-- ]; k0 + Plus @@ k /@ Range[i, nu]]; L = M = {}; a = 1; Do[ s = Smarandache[n], If[s/n < a, a = s/n; AppendTo[M, a]; AppendTo[L, n]]], {n, 40320}]; L (from EWW May 17 2004)

Cf. A002034, A094372, A094404, A094634.

Sequence in context: A105455 A083207 A145278 this_sequence A079760 A109895 A083209

Adjacent sequences: A094368 A094369 A094370 this_sequence A094372 A094373 A094374

nonn

Jonathan Sondow (jsondow(AT)alumni.princeton.edu), Apr 28 2004

More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), May 15 2004