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%I #12 Jun 20 2013 19:00:18
%S 1,2,3,4,6,8,12,40,15,16,24,30,40,45,48,60,72,80,90,120,144,180,240,
%T 252,280,360,420,504,560,720,840,1008,1260,1344,1440,1680,1920,2016,
%U 2520,2688,2880,3360,3456,4032,5040,5760,6720,8064,8640,10080,10368,11340,13440,72576,15120,17280,20160,22680,24192,25920,26880,30240
%N Denominators of incrementally smallest ratios A002034(n)/n.
%C Numerators are in A094404. Same as A094371(n)/gcd(A094371(n), A002034(A094371(n))). The factorials appear to form a subsequence.
%H J. Sondow and E. W. Weisstein, <a href="http://mathworld.wolfram.com/SmarandacheFunction.html">MathWorld: Smarandache Function</a>
%e a(5) = 6 because the 5th incrementally smallest ratio A002034(n)/n is 4/24 = 1/6.
%t (A002034[n_] := (m = 1; While[ !IntegerQ[m!/n], m++ ]; m); M = {}; Do[With[{s = A002034[n]}, If[s/n < Min[M], M = Append[M, s/n]]], {n, 120}]; Denominator[M])
%t A002034[1] := 1; A002034[n_] := Max[A002034 @@@ FactorInteger[n]]; A002034[p_, 1] := p; A002034[p_, alpha_] := A002034[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]]; M = {}; a = 2; Do[ s = A002034[n]; If[s/n < a, a = s/n; AppendTo[M, a]]], {n, 2, 40320}]; Denominator[M] (from _Eric W. Weisstein_, May 17 2004)
%Y Cf. A002034, A094371, A094404, A094634.
%K nonn,frac
%O 1,2
%A _Jonathan Sondow_, Apr 28 2004
%E More terms from _Robert G. Wilson v_, May 15 2004
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