Fickle sequences
Sorts of sequences
There are sequences of which the terms can be found by a formula.
In this way the formule t(n) = n2 generates the sequence 1 , 4 , 9 , 16 , … for natural numbers > 0.
The arithmetic progressions and geometric sequences are famous.
The harmonic sequence is less known.
In some sequences the terms can be found iteratively..
An example: t(n + 1) = t(n ) + 5 with t(1) = 3 generates the sequence 3 , 8 , 13 , 18 , 23 , …
Another example, often mentioned, is the renowned Fibonacci sequence.
Loop ending sequences repeat after some terms: a b c d e f d e f d e f ... .
See the special about it.
And then the
Fickle sequences
A fickle sequence is a subset of the sequence of natural numbers with an unpredictable course.
It is impossible to find a formula or iteration to predict a next term from the previous terms.
However it is possible to generate new terms anyhow.
Every fickle sequence contains terms with a specific property.
The anti fickle sequence contains terms within the set of natural numbers
without the specific property.
The anti fickle sequence of the prime number set is the fickle sequence
0 , 1 , 4 , 6 , 8 , 9 , 10 , 12 , 14 , 15 , 16 , 18 , 20 , 21 , 22 , … .
Each sequence has an anti sequence.
The union of a sequence and its anti sequence is the set of natural numbers.
The intersection of a sequence and its anti sequence is disjoint (empty); there are no common terms.
The (anti) fickle sequences concern a rather unknown area in mathematics.
Examples for fickle sequences
Prime numbers
A prime (number) has exactly two different dividers. See ↑ .
Asqua
By definition an asqua (a fantasy word) is a sum of two squares.
The asqua sequence contains asquas:
2 = 1² + 1² , 5 = 2² + 1² , 8 = 2² + 2² , 10 = 3² + 1² , 13 = 3² + 2² , 17 = 4² + 1² ,
18 = 3² + 3² , 20 = 4² + 2² , 25 = 4² + 3² , 26 = 5² + 1² , 29 = 5² + 2² , 32 = 4² + 4² , ... .
0² is never used.
The added anti sequence is 0 1 3 4 6 7 9 11 12 14 15 16 19 21 22 23 24, … , also fickle.
Multiplicative wisps
5! means 1 × 2 × 3 × 4 × 5 and is called five factorial. Its value is 120.
An agreement only for this site: 3!7 means 3 × 4 × 5 × 6 × 7 = 2520.
(It follows that 5! = 1!5)
These are two examples of terms concerning multiplicative wisps.
The sequence of m!n (with m < n)is a fickle sequence.
A multiplicative wisp is a multiplication of at least two sequential natural numbers.
Zero and one are excluded for an obvious reason!
The number of multiplied numbers is called the length, displayed as L.
3 × 4 × 5 × 6 × 7 = (1 × 2 × 3 × 4 × 5 × 6 × 7) ÷ (1 × 2) = 7! ÷ 2! ( = 5040 ÷ 2 = 2520 )
This wisp contains 7-2 factors. Consequently L = 5.
Can every natural number > 2 be expressed as a multiplicative wisp?
Anyhow, it is impossible for prime numbers and powers.
als numbers as 2 × 7 give problems. Then the next question will be:
which subset can be displayed as multiplicative wisps?
A first subset of the multiplicative wisps is 2 × 3 = 6 , 3 × 4 = 12 , 4 × 5 = 20 , ... .
The general term in the sequence is n(n + 1).
A second subset in the multiplicative wisps is 2 × 3 × 4 = 24 , 3 × 4 × 5 = 60 , 4 × 5 × 6 = 120 , ... .
The general term in this subsequence is n(n + 1)(n + 2).
The general formula of a term in a multiplicative wisp is t = a! b = b! ÷ (a - 1)!.
There is no general solution method to find the next term t.
The only way to find out is to factorize t and then to puzzle, wether t can be a multiplicative wisp.
Not unambiguous
120 = 4!6 = 4 × 5 × 6 and 120 = 2!5 = 2 × 3 × 4 × 5.
120 is the smallest number that can be displayed as two multiplicative wisps.
The next number with this propery is 210.
The complete set of these multiplicative wisps begins as follows:
2 × 3 = 6 , 3 × 4 = 12 , 4 × 5 = 20 , 2 × 3 × 4 = 24, 5 × 6 = 30 , 6 × 7 = 42, 7 × 8 = 56 , 3 × 4 × 5 = 60 ,
8 × 9 = 72 , 9 × 10 = 90 , 10 × 11 = 110 , 4 × 5 × 6 = 2 × 3 × 4 × 5 = 120 , 11 × 12 = 132 , 12 × 13 = 156 , ... .
It is impossible to calculate the next term out of the previous terms: a real fickle sequence.
More examples of fickle sequences
- Asquas with the two numbers to be quadrated being primes.
- Terms are asquas with the two numbers to be quadrated being asquas.
- Terms are asquas with the two numbers to be quadrated being primes.
- Terms are asquas with the two numbers to be quadrated being triangular numbers.
- Terms are binomial coefficients n over k (in Pascal's triangle) with conditions n > k and 2 < k < 2n.
- Terms are numbers of which the sum of their digits is divisible by seven.
- Terms are numbers having eight different divisors.
- Terms are numbers having eight different prime divisors.
- Terms are the product of exactly five primes.
- Terms are 7-fold of which the second digit is a 3.
- Sum of the divisors. An example: choose term number one as 12.
Determine the divisors of 12. They are 1 , 2 , 3 , 4 , 6 and 12.
Do not consider the greatest divisor, 12.
Add the divisors. The sum equals 16. This is term number two.
Continue this way. It gives the sequence 12 , 16 , 15 , 9 , 4 ending at prime 3 and 1.
Consequently L = 7.
- Terms are the sum of three squares.
It is challenging to find fundamentally different fickle sequences and
to look for further properties of new fickle sequences.
Initial ideas to research can be gained from the study of the primes.
Obvious questions are:
Is there a hidden possibility for a formula?
What is the number of terms of the sequence and what is the greatest term?
Are the differences between successive terms increasing or do they fluctuate?
Does each fickle sequence possess an anti sequence?
Is each anti sequence a fickle sequence?
Has the fickle sequence any praktical meaning?