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Article: On the numerical function [S.sup.-1.sub.min].
- Article from:
- Smarandache Notions Journal
- Article date:
- January 1, 2000
- Author:
CopyrightCOPYRIGHT 2000 American Research Press. This material is published under license from the publisher through the Gale Group, Farmington Hills, Michigan. All inquiries regarding rights should be directed to the Gale Group. (Hide copyright information)
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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!) - d((n - 1)!) where d is the number of divisors of n.
If x is a prime number, then ...