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On the numerical function [S.sup.-1.sub.min].
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In [1] on defines [S.sup.-1.sub.min] : N \ {1} [right arrow] N, [S.sup.-1.sub.min] (x) = min {[S.sup.-1](x)}, where [S.sup.-1] (x) = {a [member of] N | S(a) = x} , and S is the Smarandache function. For example [S.sup.-1](6) = {[2.sup.4], [2.sup.4] * 3, [2.sup.4] * [3.sup.2], [3.sup.2], [3.sup.2] * 2, [3.sup.2] * [2.sup.2], [3.sup.2] * [2.sup.3], [2.sup.4] * 3 * 5, [2.sup.3] * [3.sup.2] * 5, [2.sup.4] * [3.sup.2] * 5, [3.sup.2] * 5, [2.sup.4] * 5, [3.sup.2] * 5, [3.sup.2] * [2.sup.4]} and [S.sup.-1.sub.min] (6) = [3.sup.2]. If S (x) = n one knows that card ([S.sup.-1] (n)) = d(n!) ...
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