ABSTRACTS AND CONJECTURES ON SMARANDACHE NOTIONS
1) Palindromic Numbers and Iterations of the Pseudo-Smarandache Function
A number is called palindromic if it reads the same forwards and
backwards. For examples: 121, 34566543. 1111.
The Pseudo-Smarandache Function Z(n) is defined for any n >= 1 as the
smallest integer m such that n evenly divides 1 + 2 + ... + m.
There are some palindromic numbers n such that Z(n) is also palindromic:
Z(909) = 404, Z(2222) = 1111.
k 0
Let Z (n) = Z(Z(Z(...(n)))), where the function Z is executed k times. Z (n)
is, by convention, n.
k
Unsolved Question: What is the largest value of m so that for some n, Z (n)
is a palindrome for all k = 0, 1, 2, ..., m? k
Conjecture: There is no largest value of m such that for some n, Z (n) is a
palindrome for all k = 0, 1, 2, ..., m.
(C. Ashbacher)
Reference:
[1] Kashihara, Kenichiro, "Comments and Topics on Smarandache Notions and
Problems", Erhus Univ. Press, Vail, USA, 1996.
2) Computational Aspect of the Smarandache's Function
This note presents an algorithm, which tries to avoid the
factorials, for the Smarandache's function computation. The complexity of
the algorithm is studied using the main properties of the function. An
interesting inequality is found giving the complexity of the function on
the set {1, 2, ..., n}.
3) Some Upper Bounds for the Smarandache Function Average
_
Let S = (1/n) {S(1) + S(2) + ... + S(n)} be the Smarandache Function Average.
We prove that _
S <= (3/8)n + 1/4 + 2/n, for n > 5;
_
S <= (21/72)n + 1/12 - 2/n, for n > 23;
and we conjecture that
_
S <= (2n)/(ln n), for n > 1,
that we have checked with a C program for all numbers <= 10000.
(S. Tabirca, T. Tabirca)
4) Another Conjecture on Prime Numbers
One proves that the Smarandache Reverse Sequence
1, 21, 321, 4321, 54321, 654321, 7654321, ...
(obtained by concatenation of natural numbers in a decreasing order) doesn't
contains infinitely prime terms.
5) A conjecture on Smarandache Anti-Symmetric Sequence
In this paper we study a conjecture which states
the Smarandache Anti-Symmetric Sequence:
______________
123...n123...n, for n >= 1,
has no perfect-power term.
6) Analytical Formulas of 6 Smarandache Series and Their Application in
the Magic Square Theory
We present a set of analytical formulas for the computation of
the general term in each of the following sequences:
1, 12, 123, 1234, 12345, 123456, ... (smarandache consecutive sequence)
1, 11, 121, 1221. 12321, 123321, ... (smarandache symmetric sequence)
1, 212, 32123, 4321234, 543212345, ... (smarandache mirror sequence)
1, 23, 456, 7891, 23456, 789123, 4567891, ... (smarandache deconstructive
sequence)
1, 12, 21, 123, 231, 312, 1234, 2341, 3412, 4123, ... (smarandache circular
sequence)
12, 1342, 135642, 13578642, 13579108642, ... (smarandache permutation
sequence)
in order to construct 3x3 Magic Squares from k-truncated Smarandache terms.
{Paper presented to the FIRST INTERNATIONAL CONFERENCE ON SMARANDACHE TYPE
NOTIONS IN NUMBER THEORY, University of Craiova, Romania, August 21-24, 1997}
(Y. Chebrakov, V. Shmagin)
Reference:
[1] C.Dumitrescu & V.Seleacu, "Some Notions and Questions in Number Theory",
Erhus Univ. Press, Glendale, 1994.
7) The System-Graphical Analysis of Some Numerical Smarandache Sequences
An analytical investigation of 6 Smarandache Sequences of 1st kind:
1, 12, 123, 1234, 12345, 123456, ...
1, 11, 121, 1221. 12321, 123321, ...
1, 212, 32123, 4321234, 543212345, ...
1, 23, 456, 7891, 23456, 789123, 4567891, ...
1, 12, 21, 123, 231, 312, 1234, 2341, 3412, 4123, ...
12, 1342, 135642, 13578642, 13579108642, ...
permitted to state that the terms of these sequences are given by the
general recurrent expression:
psi(a )
n
a = sigma(a 10 + a + 1), where phi(n) and psi(n) are
phi(n) n n
functions, sigma is an operator.
The main goal of the present research is to demonstrate that the system-
graphical analysis results of these sequences possess big aesthetic,
cognitive and applied significances.
All Smarandache circumferences associated with these sequences reveal
Magic properties, hence it will be very interesting to confront them
with the ancient Chinese hexagrams.
(Y. Chebrakov, V. Shmagin)
8) Smarandache Prime-Digital Sub-Sequence (IV)
"Personal Computer World" Numbers Count of February 1997
presented some of the Smarandache Sequences and related open
problems.
One of them defines the Smarandache Prime-Digital Sub-Sequence
as the ordered set of primes whose digits are all primes:
2, 3, 5, 7, 23, 37, 53, 73, 223, 227, 233, 257, 277, ... .
We used a computer program in Ubasic to calculate the first 100
terms of the sequence. The 100-th term is 33223.
Sylvester Smith [1] conjectured that the sequence is infinite. In
this paper we will prove that this sequence is in fact infinite.
(H. Ibstedt)
Reference:
[1] Smith, Sylvester, "A Set of Conjectures on Smarandache
Sequences", in ,
India, Vol. 15E, No. 1, 1996, pp. 101-107.
9) Smarandache G Add-On Sequence (V)
"Personal Computer World" Numbers Count of February 1997
presented some of the Smarandache Sequences and related open
problems.
Let G = {g1, g2, ..., gk, ...} be an ordered set of positive
integers with a given property G.
Then the corresponding Smarandache G Add-On Sequence is defined
through
1+log (g )
10 k
SG = {a : a = g , a = a 10 + g , k >= 1}.
i 1 1 k k-1 k
I study some particular cases of this sequence, that I have
presented at the FIRST INTERNATIONAL CONFERENCE ON SMARANDACHE
TYPE NOTIONS IN NUMBER THEORY, University of Craiova, Romania,
August 21-24, 1997.
(H. Ibstedt)
10) Examples of Smarandache G Add-On Sequences (VI)
The following particular cases are studied:
1) Smarandache Odd Sequence is generated by choosing
G = {1, 3, 5, 7, 9, 11, ...}, and it is:
1, 13, 135, 1357, 13579, 1357911, 13571113, ... .
Using the elliptic curve prime factorization program we find
the first five prime numbers among the first 200 terms of this
sequence, i.e. the ranks 2, 15, 27, 63, 93.
But are they infinitely or finitely many?
2) Smarandache Even Sequence is generated by choosing
G = {2, 4, 6, 8, 10, 12, ...}, and it is:
2, 24, 246, 2468, 246810, 24681012, ... .
Searching the first 200 terms of the sequence we didn't find
any n-th perfect power among them, no perfect square, nor even
of the form 2p, where p is a prime or pseudo-prime.
Conjecture: There is no n-th perfect power term!
3) Smarandache Prime Sequence is generated by choosing
G = {2, 3, 5, 7, 11, 13, 17, ...}, and it is:
2, 23, 235, 2357, 235711, 23571113, 2357111317, ... .
Terms #2 and #4 are primes; terms #128 (of 355 digits) and #174
(of 499 digits) might be, but we couldn't check -- among the first
200 terms of the sequence.
Question: Are there infinitely or finitely many such primes?
(H. Ibstedt)
11) Smarandache Non-Arithmetic Progression (I)
"Personal Computer World" Numbers Count of February 1997
presented some of the Smarandache Sequences and related open
problems.
One of them defines the Smarandache t-Term Non-Arithmetic Progression
as the set:
{a : a is the smallest integer such that a > a ,
i i i i-1
and there are at most t-1 terms in an arithmetic progression}.
A QBASIC program is designed to implement a strategy for
building a such progression, and a table for the 65 first terms of
the smarandache non-arithmetic progressions for t=3 to 15 is given.
(H. Ibstedt)
12) Smarandache Prime-Product Sequence (II)
"Personal Computer World" Numbers Count of February 1997
presented some of the Smarandache Sequences and related open
problems.
One of them defines the Smarandache Prime-Product Sequence
as the set:
{t : t = p # + 1, where p is the n-th prime and p # denotes
n n n n n
the product of all first n prime numbers}.
Question: how many of them are prime?
Using a computer program in QBASIC we find the first six prime
numbers in this sequence: 3, 7, 31, 211, 2311, 200560490131.
Are there infinitely many?
The number of primes q among the first 200 terms is 6 <= q <= 9.
The three terms which are either primes or pseudo primes are terms
numero 75, 171, and 172 having 154, 425, and 428 digits respectively.
The later two are generated by the prime twins 1019 and 1021.
(H. Ibstedt)
13) Smarandache Square-Factorial Sequence (III)
"Personal Computer World" Numbers Count of February 1997
presented some of the Smarandache Sequences and related open
problems.
One of them defines the Smarandache Square-Factorial Sequence
as the set:
2
{f : f = (n!) + 1}
n n
We study how many terms of this sequence are prime? Among the 40
first terms we got the following primes: 2, 5, 37, 577, 14401,
131681894401, 13168189440001, 1593350922240001, 38775788043632640001,
384956219213331276939737002152967117209600000001.
Are there infinitely many such primes?
the product of all first n prime numbers.
Question: how many of them are prime?
Using a computer program in QBASIC we find six prime numbers in
this sequence: 3, 7, 31, 211, 2311, 200560490131. Are there
infinitely many?
The number of primes q among the first 200 terms is 6 <= q <= 9.
The three terms which are either primes or pseudo primes are terms
numero 75, 171, and 172 having 154, 425, and 428 digits respectively.
The later two are generated by the prime twins 1019 a
(H. Ibstedt)
14) Smarandache Concatenation Type Sequences
Let s , s , s , ..., s , ... be an infinite integer sequence
1 2 3 n
(noted by S). Then the Smarandache Concatenation is defined as:
____ ______
s , s s , s s s , ... .
1 1 2 1 2 3
I search, in some particular cases, how many terms of this
concatenated S-sequence belong to the initial S-sequence.
15) Smarandache Partition Type Sequences
Let f be an arithmetic function, and R a k-relation among numbers.
How many times can n be expressed under the form of;
n = R ( f(n ), f(n ), ..., f(n )),
1 2 k
for some k and n , n , ..., n such that n + n + ... + n = n ?
1 2 k 1 2 k
Look at some particular cases: How many times can be n express as a
sum of non-null squares (or cubes, or m-powers)?
16) A conjecture on Smarandache Anti-Symmetric Sequence
In this paper we study a conjecture which states
the Smarandache Anti-Symmetric Sequence:
______________
123...n123...n, for n >= 1,
has no perfect-power term.
17) On Smarandache Deducibility Theorem
This theorem is defined in the Propositional Calculus of
Mathematics Logic as:
___
If I--- A ___I B , for i = 1, 2, ..., n, then:
i i
___
a) I--- A ^ A ^ ... ^ A ___I B ^ B ^ ... ^ B ;
1 2 n 1 2 n
___
a) I--- A v A v ... v A ___I B v B v ... v B .
1 2 n 1 2 n
We study a similar theorem in the case when the logic operators "^"
and "v" are replaced by Sheffer's operator.
Reference:
F. Smarandache, "Deducibility theorems in mathematics logics", in , seria St. Matematice, Vol. XVII, fasc. 2, 1979,
pp. 163-168.
We particularize S anf f to study interesting cases of this type of
sequences.
18) On Smarandache Concurrent Lines
If a polygon with n sides (n >= 4) is circumscribed to a circle,
then there are at least three concurrent lines among the polygon's diagonals
and the lines which join tangential points of two non-adjacent sides.
(This is known as Smarandache Concurrent Lines, and generalizes a geometric
theorem of Newton.)
In this paper we try to extend this result in a three-dimensional space.
Reference:
F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress,
Fes, Morocco, 1983, Problem & Solution # 5.36, p. 54.
We particularize S anf f to study interesting cases of this type of
sequences.
19) On Smarandache Cevians Theorem
Let AA', BB', CC' be three concurrent cevians (lines), in the
point P, in the triangle ABC. Then:
PA/PA' + PB/PB' + PC/PC' >= 6,
and
PA PB PC BA CB AC
---- . ---- . ---- = ---- . ---- . ---- >= 8
PA' PB' PC' BA' CB' AC'
(Smarandache Cevians Theorem).
In this paper we extend this result for a quadrilateral.
Reference:
F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress,
Fes, Morocco, 1983, Problems & Solutions # 5.37, p. 55, # 5.40, p. 58.
We particularize S anf f to study interesting cases of this type of
sequences.
20) On Smarandache Podaire Theorem
Let AA', BB', CC' be the altitudes of the triangle ABC.
Thus A'B'C' is the podaire triangle of the triangle ABC.
Note AB = c, BC = a, CA = b, and A'B' = c', B'C' = a', C'A' = b'.
Then:
a'b' + b'c' + c'a' <= 1/4 (a^2 + b^2 + c^2)
(Smarandache Podaire Theorem).
In this paper we study this result for a quadrilateral.
Reference:
F. Smarandache, "Problemes avec and sans problemes!", Ed. Somipress,
Fes, Morocco, 1983, Problem & Solution # 5.41, p. 59.
We particularize S anf f to study interesting cases of this type of
sequences.
21) On Smarandache Type Bases
Considering any number wriiten in Smarandache Prime/Square/Cubic/
General Base, as written in the decimal base, we check:
a) how many of them are primes?
b) how many of them are perfect powers
(particularly: perfect squares, or perfect cubes)?
22) Are There Finitely or Infinitely Smarandache Lucky Numbers?
A number is said to be a Smarandache Lucky Number if an incorrect
calculation leads to a correct result.
For example, in the fraction 64/16 if the 6's are incorrectly cancelled
(simplified) the result 4 is still correct. (We exclude trivial examples of
the form 600/200 where non-aligned zeros are cancelled.)
Is the set of all fractions, where such (or onather) incorrect calculation
leads to a correct result, finite or infinite?
More general: The Smarandache Lucky Method/Algorithm/Operation/etc. is
said to be any incorrect method or algorithm or operation etc. which leads to
a correct result. The wrong calculation should be fun, somehow similarly
to the students' common mistakes, or to produce confusions or psradoxes.
Can someone give an example of a Smarandache Lucky Derivation, or
Integration, or Solution to a Differential Equation?
(C. Ashbacher)
Reference:
[1] Smarandache, Florentin, "Collected Papers" (Vol. II), University of
Kishinev, 1997.
22) Solved and Unsolved Problems on Pseudo-Smarandache Function
A) In this note are solved the Problems:
1) Let p be a positive prime and s be an integer >= 2.
Then: Z(p^s) = p^(s+1)-1, if p is even; or p^s-1, if p is odd.
2) The solution set of the diophantine equation Z(x) = 8 is {9, 12, 18, 36}.
3) For any positive integer n the diophantine equation Z(x) = n has
solutions.
B) Unsolved Problems:
4) The diophantine equation Z(x) = Z(x+1) has no solutions.
5) For any given positive number r there exists an integer s so that the
absolute value of Z(s) - Z(s+1) is greater than r.
(Where Z(n) is the Pseudo-Smarandache Function: the smallest integer m
such that n evenly divides 1+2+3+...+m.)
Reference:
[1] Kashihara, Kenichiro, "Comments and Topics on Smarandache Notions and
Problems", Erhus Univ. Press, Vail, 1996.
23) Construction of Elements of the Smarandache Square-Partial-Digital
Subsequence
The Smarandache Square-Partial-Digital Subsequence (SSPDS) is the
sequence of square integers which admit a partition for which each segment is
a square integer. An example is 506^2 = 256036, which has partition 256/0/36.
C. Ashbacher showed that SSPDS is infinite by exibiting two infinite families
of elements. We will extend his results by showing how to construct infinite
families of elements of SSPDS containing desired patterns of digits.
Unsolved Question 1:
441 belongs to SSPDS, and his square 441^2 = 194481 also belongs to SSPDS.
Can an example be found of integers m, m^2, m^4 all belonging to SSPDS?
Unsolved Question 2:
It is relatively easy to find two consecutive squares in SSDPS, i.e.
12^2 = 144 and 13^2 = 169.
Does SSDPS also contain three or more consecutive squares?
What is the maximum length?
(L. Widmer)
24) Perfect Powers in Smarandache Type Expressions (I)
How many primes are there in the Smarandache Expression:
x^y + y^x,
where gcd(x, y) = 1 ? [J. Castillo & P. Castini]
K. Kashihara announced that there are only finitely many numbers of the
above form which are products of factorials.
In this note we propose the following conjecture:
Let a, b, and c three integers with ab nonzero. Then the equation:
ax^y + by^x = cz^n, with x, y, n >= 2, and gcd(x, y) = 1,
has finitely many solutions (x, y, z, n).
And we prove some particular cases of it.
25) Products of Factorials in Smarandache Type Expressions (II)
J. Castillo ["Mathematical Spectrum", Vol. 29, 1997/8, 21] asked
how many primes are there in the Smarandache n-Expression:
x1^x2 + x2^x3 + ... + xn^x1,
where n > 1, x1, x2, ..., xn > 1, and gcd (x1, x2, ..., xn) = 1 ?
[This is a generalization of the Smarandache 2-Expression: x^y + y^x.]
In this note we announce a lower bound for the size of the largest prime
divisor of an expression of type ax^y + by^x, where ab is nonzero,
x, y >= 2, and gcd (x, y) = 1.
(F. Luca)
26) The Smarandache General Periodic Sequence
Definition:
Let S be a finite set, and f : S ---> S be a function defined
for all elements of S.
There will always be a periodic sequence whenever we repeat the composition
of the function f with itself more times than card(S), accordingly to the
box principle of Dirichlet.
[The invariant sequence is considered a periodic sequence whose period length
has one term.]
Thus the Smarandache General Periodic Sequence is defined as:
a1 = f(s), where s is an element of S;
a2 = f(a1) = f(f(s));
a3 = f(a2) = f(f(a1)) = f(f(f(s)));
and so on.
We particularize S anf f to study interesting cases of this type of
sequences.
(M. R. Popov)
27) The Two-Digit Smarandache Periodic Sequence (I)
Let N1 be an integer of at most two digits and let N1' be its
digital reverse. One defines the absolute value N2 = abs (N1 - N1').
And so on: N3 = abs (N2 - N2'), etc. If a number N has one digit only,
one considers its reverse as Nx10 (for example: 5, which is 05, reversed will
be 50). This sequence is periodic.
Except the case when the two digits are equal, and the sequence becomes:
N1, 0, 0, 0, ...
the iteration always produces a loop of length 5, which starts on the second
or the third term of the sequence, and the period is 9, 81, 63, 27, 45
or a cyclic permutation thereof.
(H. Ibstedt)
Reference:
[1] Popov, M.R., "Smarandache's Periodic Sequences", in , University of Sheffield, U.K., Vol. 29, No. 1, 1996/7, p. 15.
28) The n-Digit Smarandache Periodic Sequence (II)
Let N1 be an integer of at most n digits and let N1' be its
digital reverse. One defines the absolute value N2 = abs (N1 - N1').
And so on: N3 = abs (N2 - N2'), etc. If a number N has less than n digits,
one considers its reverse as N'x(10^k), where N' is the reverse of N and
k is the number of missing digits, (for example: the number 24 doesn't have
five digits, but can be written as 00024, and reversed will be 42000).
This sequence is periodic according to Dirichlet's box principle.
The Smarandache 3-Digit Periodic Sequence (domain 100 <= N1 <= 999):
- there are 90 symmetric integers, 101, 111, 121, ..., for which N2 = 0;
- all other initial integers iterate into various entry points of the same
periodic subsequence (or a cyclic permutation thereof) of five terms:
99, 891, 693, 297, 495.
The Smarandache 4-Digit Periodic Sequence (domain 1000<= N1 <= 9999):
- the largest number of iterations carried out in order to reach the first
member of the loop is 18, and it happens for N1 = 1019;
- iterations of 8818 integers result in one of the following loops (or a
cyclic permutation thereof): 2178, 6534; or 90, 810, 630, 270, 450; or
909, 8181, 6363, 2727, 4545; or 999, 8991, 6993, 2997, 4995;
- the other iterations ended up in the invariant 0.
(H. Ibstedt)
29) The 5-Digit and 6-Digit Smarandache Periodic Sequences (III)
Let N1 be an integer of at most n digits and let N1' be its
digital reverse. One defines the absolute value N2 = abs (N1 - N1').
And so on: N3 = abs (N2 - N2'), etc. If a number N has less than n digits,
one considers its reverse as N'x(10^k), where N' is the reverse of N and
k is the number of missing digits, (for example: the number 24 doesn't have
five digits, but can be written as 00024, and reversed will be 42000).
This sequence is periodic according to Dirichlet's box principle, leading to
invariant or a loop.
The Smarandache 5-Digit Periodic Sequence (domain 10000 <= N1 <= 99999):
- there are 920 integers iterating into the invariant 0 due to symmetries;
- the other ones iterate into one of the following loops (or a cyclic
permutation of these): 21978, 65934; or 990, 8910, 6930, 2970, 4950; or
9009, 81081, 63063, 27027, 45045; or 9999, 89991, 69993, 29997, 49995.
The Smarandache 6-Digit Periodic Sequence (domain 100000 <= N1 <= 999999):
- there are 13667 integers iterating into the invariant 0 due to symmetries;
- the longest sequence of iterations before arriving at the first loop
member is 53 for N1 = 100720;
- the loops have 2, 5, 9, or 18 terms.
(H. Ibstedt)
30) The Smarandache Subtraction Periodic Sequences (IV)
Let c be a positive integer. Start with a positive integer N, and
let N' be its digital reverse. Put N1 = abs(N1' - c), and let N1' be its
digital reverse. Put N2 = abs (N1' - c), and let N2' be its digital reverse.
And so on. We shall eventually obtain a repetition.
For example, with c = 1 and N = 52 we obtain the sequence: 52, 24, 41, 13,
30, 02, 19, 90, 08, 79, 96, 68, 85, 57, 74, 46, 63, 35, 52, ... . Here a
repetition occurs after 18 steps, and the length of the repeating cycle is 18.
First example: c = 1, 10<= N <= 999.
Every other member of this interval is an entry point into one of five cyclic
periodic sequences (four of these are of length 18, and one of length 9).
When N is of the form 11k or 11k-1, then the iteration process results in 0.
Second example: 1 <= c <= 9, 100 <= N <= 999.
For c = 1, 2, or 5 all iterations result in the invariant 0 after, sometimes,
a large number of iterations.
For the other values of c there are only eight different possible values for
the length of the loops, namely 11, 22, 33, 50, 100, 167, 189, 200.
For c = 7 and N = 109 we have an example of the longest loop obtained: it
has 200 elements, and the loop is closed after 286 iterations.
(H. Ibstedt)
31) The Smarandache Multiplication Periodic Sequences (V)
Let c > 1 be a positive integer. Start with a positive integer N,
multiply each digit x of N by c and replace that digit by the last digit of
cx to give N1. And so on. We shall eventually obtain a repetition.
For example, with c = 7 and N = 68 we obtain the sequence:
68, 26, 42, 84, 68, ... .
Integers whose digits are all equal to 5 are invariant under the given
operation after one iteration.
One studies the Smarandache One-Digit Multiplication Periodic Sequences only.
(For c of two or more digits the problem becomes more complicated.)
If c = 2, there are four term loops, starting on the first or second term.
If c = 3, there are four term loops, starting with the first term.
If c = 4, there are two term loops, starting on the first or second term
(could be called Smarandache Switch or Pendulum).
If c = 5 or 6, the sequence is invariant after one iteration.
If c = 7, there are four term loops, starting with the first term.
If c = 8, there are four term loops, starting with the second term.
If c = 9, there are two term loops, starting with the first term (pendulum).
(H. Ibstedt)
32) The Smarandache Mixed Composition Periodic Sequences (VI)
Let N be a two-digit number. Add the digits, and add them again if
the sum is greater than 10. Also take the absolute value of their difference.
These are the first and second digits of N1. Now repeat this.
For example, with N = 75 we obtain the sequence: 75, 32, 51, 64, 12, 31, 42,
62, 84, 34, 71, 86, 52, 73, 14, 53, 82, 16, 75, ... .
There are no invariants in this case. Four numbers: 36, 90, 93, and 99
produce two-element loops. The longest loops have 18 elements. There also
are loops of 4, 6, and 12 elements.
There will always be a periodic (invariant) sequence whenever we have a
function f : S ---> S, where S is a finite set,
and we repeat the function f more times than card(S).
Thus the Smarandache General Periodic Sequence is defined as:
a1 = f(s), where s is an element of S;
a2 = f(a1) = f(f(s));
a3 = f(a2) = f(f(a1)) = f(f(f(s)));
and so on.
(H. Ibstedt)
33) New Smarandache Sequences: The Family of Metallic Means
The family of Smarandache Metallic Means (whom most prominent
members are the Golden Mean, Silver Mean, Bronze Mean, Nickel Mean, Copper
Mean, etc.) comprises every quadratic irrational number that is the
positive solution of one of the algebraic equations
2 2
x - nx - 1 = 0 or x - x - n = 0,
where n is a natural number.
All of them are closely related to quasi-periodic dynamics, being therefore
important basis of musical and architectural proportions. Through the
analysis of their common mathematical properties, it becomes evident that
they interconnect different human fields of knowledge, in the sense defined
by Florentin Smarandache ("Paradoxist Mathematics").
Being irrational numbers, in applications to different scientific
disciplines, they have to be approximated by ratios of integers -- which is
the goal of this paper.
(Vera W. de Spinadel)
34) About the Behaviour of Some New Functions in the Number Theory
We investigate and prove the functions:
S1 : N-{0,1} ---> N, S1(n) = 1/S(n);
S2 : N* ---> N, S2(n) = S(n)/n
verify the Lipschitz condition, but the functions:
S3 : N-{0,1} ---> N, S3(n) = n/S(n);
Fs : N* ---> N, x
Fs(x) = sigma( S(p ) for i from 1 to pi(x),
i
where p are the prime numbers not greater than x and
i
pi(x) is the number of them;
Theta : N* ---> N, x
Theta(x) = sigma S(p ), where p are prime numbers
i i
which divide x;
_____
Theta : N* ---> N,
_____ x
Theta(x) = sigma S(p ), where p are prime numbers
i i
which do not divide x;
where S(n) is the Smarandache function for all six previous functions,
verify the Lipschitz condition.
(V. Seleacu, S. Zanfir)