A094372 Denominators of incrementally smallest ratios A002034(n)/n.

1, 2, 3, 4, 6, 8, 12, 40, 15, 16, 24, 30, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 252, 280, 360, 420, 504, 560, 720, 840, 1008, 1260, 1344, 1440, 1680, 1920, 2016, 2520, 2688, 2880, 3360, 3456, 4032, 5040, 5760, 6720, 8064, 8640, 10080, 10368, 11340, 13440, 72576, 15120, 17280, 20160, 22680, 24192, 25920, 26880, 30240

Offset: 1

Author

Jonathan Sondow, Apr 28 2004

Keywords

• nonn
• frac

Comments

Numerators are in A094404. Same as A094371(n)/gcd(A094371(n), A002034(A094371(n))). The factorials appear to form a subsequence.

Examples

			a(5) = 6 because the 5th incrementally smallest ratio A002034(n)/n is 4/24 = 1/6.
		

Links

• J. Sondow and E. W. Weisstein, MathWorld: Smarandache Function

Crossrefs

Cf. A002034, A094371, A094404, A094634.

Programs

Mathematica

(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])
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 = Infinity; Do[ s = A002034[n]; If[s/n < a, a = s/n; AppendTo[M, a]], {n, 2, 40320}]; Denominator[M] (* Eric W. Weisstein, May 17 2004 *)

Extensions

More terms from Robert G. Wilson v, May 15 2004