FUNCTIONAL SMARANDACHE ITERATIONS
1) Functional Smarandache Iteration of First Kind:
Let f: A ---> A be a function, such that f(x) <= x for all x,
and
min {f(x), x belongs to A} > = m0, different from negative infinity.
Let f have p >= 1 fix points: m0 <= x1 < x2 < ... < xp.
[The point x is called fix if f(x) = x.]
Then
SI1 (x) = the smallest number of iterations k such that
f
f(f(...f(x)...)) = constant.
iterated k times
Example:
Let n > 1 be an integer, and d(n) be the number of positive divisors of n,
d: N ---> N.
Then SI1 (n) is the smallest number of iterations k
d
such that d(d(...d(n)...)) = 2;
iterated k times
because d(n) < n for n > 2, and the fix points of the function d are 1
and 2.
Thus SI1 (6) = 3, because d(d(d(6))) = d(d(4)) = d(3) = 2 = constant.
d
SI1 (5) = 1, because d(5) = 2.
d
2) Functional Smarandache Iteration of Second Kind:
Let g: A ---> A be a function, such that g(x) > x for all x,
and let b > x. Then:
SI2 (x, b) = the smallest number of iterations k such that
g
g(g(...g(x)...)) >= b.
iterated k times
Example:
Let n > 1 be an integer, and sigma(n) be the sum of positive divisors
of n (1 and n included), sigma: N ---> N.
Then SI2 (n, b) is the smallest number of iterations k such that
sigma
sigma(sigma(...sigma(n)...)) >= b,
iterated k times
because sigma(n) > n for n > 1.
Thus SI2 (4, 11) = 3, because sigma(sigma(sigma(4))) =
sigma
sigma(sigma(7)) = sigma(8) = 15 >= 11.
3) Functional Smarandache Iteration of Third Kind:
Let h: A ---> A be a function, such that h(x) < x for all x,
and let b < x. Then:
SI3 (x, b) = the smallest number of iterations k such that
h
h(h(...h(x)...)) <= b.
iterated k times
Example:
Let n be an integer and gd(n) be the greatest divisor of n, less than n,
gd: N* ---> N*. Then gd(n) < n for n > 1.
SI3 (60, 3) = 4, because gd(gd(gd(gd(60)))) = gd(gd(gd(30))) =
gd
gd(gd(15)) = gd(5) = 1 <= 3.
References:
[1] Ibstedt, H., "Smarandache Iterations of First and Second Kinds",
,
Vol. 17, No. 4, Issue 106, 1996, p. 680.
[2] Ibstedt, H., "Surfing on the Ocean of Numbers - A Few Smarandache
Notions and Similar Topics", Erhus University Press, Vail, 1997;
pp. 52-58.
[3] Smarandache, F., "Unsolved Problem: 52", , Xiquan Publishing House, Phoenix, 1993.