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Decomposition into weight * level + jump
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Decomposition of numbers into weight × level + jump
The main goal of this page is to provide a new way to see certain numbers by their decomposition into weight × level + jump. This decomposition can be seen as a generalization of the sieve of Eratosthenes (which is the particular case of the sequence of natural numbers). Applied to prime numbers, this decomposition is used to obtain a new classification of primes.
Author: Rémi Eismann
Principles
Principle of decomposition: we choose the smallest weight such that in the Euclidean division of a number by its weight, the remainder is the jump (first difference, gap). The quotient will be the level.
Principles of classification: if the number is not decomposable, it is not classified. If the weight is greater than the level then the number is classified by level, if not then it is classified by weight.
Definitions
Let be a strictly increasing integer sequence.
Jump
The jump (first difference, gap) of is defined as
l(n)
l(n) is defined as
Alternative definition with mod function
Weight
The weight of is defined as
Alternative definition with mod function
Level
The level of is defined as
Decomposition criterion
A strictly increasing integer sequence, can be decomposed into weight × level + jump when is different from 0 which can be rewritten when:
or
or
Unique decomposition
The weight is the smallest such that in the Euclidean division of by its weight , the quotient is the level , and the remainder is the jump . We have the unique decomposition
Principles of classification
- If for ,
then is not classified.
- If for ,
then is classified by level, if not then it is classified by weight.
Algorithms
Naive algorithm (PARI/GP):
decompnaive(n,n1)={
/*strictly increasing*/
if(n>=n1,print("n1 must be greater than n");return);
/*jump*/
d=n1-n;
/*l=n-d if n>2*d else the number is not decomposable*/
if(n>2*d,l=n-d,print(d, ", 0, 0");return);
/*we look for the weight to jump+1 until l*/
for(k=d+1,l,if(n%k==d,print(n," = ",k," * ",l/k," + ",d);return));
}
Algorithm "newSieve":
decompsieve(n,n1)={
/*strictly increasing*/
if(n>=n1,print("n1 must be greater than n");return);
/*jump*/
d=n1-n;
/*l=n-d if n>2*d else the number is not decomposable*/
if(n>2*d,l=n-d,print(d, ", 0, 0");return);
/*we look for the weight to jump+1 until sqrt(l)*/
for(k=d+1,sqrt(l),if(n%k==d,print(n," = ",k," * ",l/k," + ",d);return));
/*we look for the level to jump until 1 (--)*/
forstep(le=d,1,-1,if(n%floor(l/le)==d,print(n," = ",l/le," * ",le," + ",d);return));
}
Algorithm "newSieve" is the fastest for numbers classified by level.
General remarks
Numbers of decompositions into weight × level + jump | |
1 | 0 |
2 | 0 |
3 | 1 |
4 | 1 |
5 | 2 |
6 | 2 |
7 | 3 |
8 | 3 |
9 | 4 |
10 | 4 |
11 | 5 |
... | ... |
The numbers of decompositions possible of into weight × level + jump is A004526 (with an offset at 1).
The sequence with the the greatest growth that can be decomposed is A003312.
Throughout this page, the jump is the first difference but we can take the second, third... difference. See A133346 and A133347 for primes.
Decomposition of natural numbers, the sieve of Eratosthenes
If the decomposition is possible (i.e. if n > 2), we have:
The weight is the smallest prime factor of and the level is the largest proper divisor of . The natural numbers classified by weight are the composites + 1 and the natural numbers classified by level are the primes + 1. The decomposition into weight × level + jump of natural numbers is thus a reformulation of the sieve of Eratosthenes.
Plot of log(A020639) vs log(A032742) for n ≤ 10^4, the sieve of Eratosthenes (OEIS graph):
Decomposition of prime numbers
, and are the only primes not decomposable^{[1]}. Except for , and , the decomposition into weight × level + jump of prime numbers is:
Plot of log(A117078) vs log(A117563) for n ≤ 10^4 (OEIS graph):
A classification of primes connected to the OEIS
Primes of level (1;i)
Principle of classification for primes of level 1:
- If for p(n), l(n) is a prime says prime(n-i) then p(n) is of level (1;i).
- Primes of level (1;1) are the balanced primes: (A006562);
- Primes of level (1;2): A117876;
- Primes of level (1;3): A118467;
- ...
Direct relations
For different from 2, 3 and 7, we have:
Primes classified by weight
For primes classified by weight (Cf. A162175) (primes for which ), we have:
82,89 % of primes are classified by weight for .
We can see that by definition, the primes classified by weight follow Legendre's conjecture and Andrica's conjecture.
Primes classified by level
For primes classified by level (Cf. A162174) (primes for which ), we have:
17,11 % of primes are classified by level for .
Knowing that the primes are rarefying among the natural numbers and according to the numerical data, we make the following conjecture:
- Conjecture 9: The prime numbers classified by level are rarefying among the primes.
Lesser of twin primes
If is a lesser of twin prime greater than 3 then has a weight of 3. If has a weight of 3 then is a lesser of twin prime greater than 3^{[1]}.
Conjectures
The well-known conjecture on the existence of an infinity of twin primes can be rewritten as:
- Conjecture 1: The number of primes with a weight equal to 3 is infinite.
To extend this conjecture we make these two conjectures:
- Conjecture 2: The number of primes with a weight equal to k is infinite for any which is not a multiple of 2;
- Conjecture 3: The number of primes of level is infinite for any which is not a multiple of 2.
- Conjecture 4: Except for p(6) = 13, p(11) = 31, p(30) = 113, p(32) = 131 et p(154) = 887, primes which are classified by level have a weight which is itself a prime.
The conjecture on the existence of an infinity of balanced primes can be rewritten as:
- Conjecture 5: The number of primes of level (1;1) is infinite.
That we can easily generalize by:
- Conjecture 6: The number of primes of level (1;i) is infinite for any i ≥ 1.
Knowing that the primes are rarefying among the natural numbers and according to the numerical data, we make the following conjecture:
- Conjecture 9: The prime numbers classified by level are rarefying among the primes.
Decomposition of odd numbers
If the decomposition is possible, we have:
Plot of log(A090368) vs log(A184726) for n ≤ 10^4 (OEIS graph):
Decomposition of even numbers
If the decomposition is possible, we have:
Plot of log(A090369) vs log(A184727) for n ≤ 10^4 (OEIS graph):
Decomposition of composite numbers
If the decomposition is possible, we have:
Plot of log(A130882) vs log(A179621) for n ≤ 10^4 (OEIS graph):
Decomposition of semiprimes
If the decomposition is possible, we have:
Plot of log(A130533) vs log(A184729) for n ≤ 10^4 (OEIS graph):
Decomposition of 3-almost primes
If the decomposition is possible, we have:
Plot of log(A130650) vs log(A184753) for n ≤ 10^4 (OEIS graph):
Decomposition of lucky numbers
If the decomposition is possible, we have:
Plot of log(A130889) vs log(A184828) for n ≤ 10^4 (OEIS graph):
Decomposition of prime powers
If the decomposition is possible, we have:
Plot of log(A184829) vs log(A184831) for n ≤ 10^4 (OEIS graph):
Decomposition of squarefree numbers
If the decomposition is possible, we have:
Plot of log(A184832) vs log(A184834) for n ≤ 10^4 (OEIS graph):
Decomposition of triangular numbers
If the decomposition is possible, we have:
Plot of log(A130703) vs log(A184219) for n ≤ 10^4 (OEIS graph):
Decomposition of squares
If the decomposition is possible, we have:
Plot of log(A133150) vs log(A184221) for n ≤ 10^4 (OEIS graph):
Decomposition of pentagonal numbers
If the decomposition is possible, we have:
Plot of log(A133151) vs log(A184751) for n ≤ 10^3 (OEIS graph):
Sequences
Sequences related to the decomposition.
See also
Notes
- ↑ ^{1.0} ^{1.1} Rémi Eismann, arXiv:0711.0865 (pdf) [1]
External links
- arXiv:0711.0865: Decomposition into weight * level + jump and application to a new classification of primes, Rémi Eismann.