Decomposition 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. We see the Fundamental theorem of arithmetic and the sieve of Eratosthenes in the decomposition of natural numbers. Applied to prime numbers, this decomposition is used to obtain a new classification of primes.

arXiv:0711.0865 [math.NT]: Decomposition into weight * level + jump and application to a new classification of prime numbers, 2007 - 2010.

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 ${\{a(n)\}}_{n=i_{\min }}^{\infty }$ be a strictly increasing integer sequence.

Jump

The jump (first difference, gap) of $a(n)$ is defined as

$d(n):=a(n+1)-a(n).\,$

l(n)

l(n) is defined as

$l(n):={\begin{cases}a(n)-d(n)&{\text{if }}a(n)-d(n)>d(n),\\0{\rm {~otherwise}}.\end{cases}}\,$

Alternative definition with mod function

$l(n):={\rm {largest~}}l{\rm {~such~that~}}a(n+1)=a(n)+a(n){\rm {~}}mod{\rm {~}}l,{\rm {~or~}}0{\rm {~if~no~such~}}l{\rm {~exists.~}}$

Weight

The weight of $a(n)$ is defined as

$k(n):={\begin{cases}{\rm {smallest~}}k>d(n){~s.t.~}k|l(n),\\0{\rm {~if~}}l(n)=0.\end{cases}}\,$

Alternative definition with mod function

$k(n):={\rm {smallest~}}k{\rm {~such~that~}}a(n+1)=a(n)+a(n){\rm {~}}mod{\rm {~}}k,{\rm {~or~}}0{\rm {~if~no~such~}}k{\rm {~exists.~}}$

Level

The level of $a(n)$ is defined as

$L(n):={\begin{cases}{\frac {l(n)}{k(n)}}&{\text{if }}k(n)>0,\\0&{\text{if }}k(n)=0.\end{cases}}\,$

Decomposition criterion

A strictly increasing integer sequence, ${\{a(n)\}}_{n=i_{\min }}^{\infty }$ can be decomposed into weight × level + jump when $l(n)$ is different from 0 which can be rewritten when:

$a(n)-d(n)>d(n),\,$

or

$d(n)<{\frac {1}{2}}\times a(n),\,$

or

$a(n+1)<{\frac {3}{2}}\times a(n)\,$

Unique decomposition

The weight $k(n)$ is the smallest such that in the Euclidean division of $a(n)$ by its weight $k(n)$, the quotient is the level $L(n)$, and the remainder is the jump $d(n)$. We have the unique decomposition

$a(n)=k(n)\times L(n)+d(n)={\rm {weight}}\times {\rm {level}}+{\rm {jump}}\,$

Principles of classification

• If for $a(n)$,

$l(n)=k(n)=L(n)=0\,$

then $a(n)$ is not classified.

• If for $a(n)$,

$k(n)>L(n)$

then $a(n)$ 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

The sequence with 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

If the decomposition is possible (i.e. if $n>2$), we have:

• A000027(n) = A020639(n-1) × A032742(n-1) + 1 (l(n) = A000027(n-1))

The weight is the smallest prime factor of $n-1$ and the level is the largest proper divisor of $n-1$. The natural numbers classified by weight are the $composites+1$ and the natural numbers classified by level are the $primes+1$. Since the jump is constant, this decomposition can be reduced as the decomposition of $l(n)$ into weight × level, and by decomposing successively the level, we come back to the Fundamental theorem of arithmetic. We see the Fundamental theorem of arithmetic and the sieve of Eratosthenes on the graph.

Plot of log(A020639) vs log(A032742) for n ≤ 10^4, the sieve of Eratosthenes (OEIS graph):

Decomposition of prime numbers

$p(1)=2$, $p(2)=3$ and $p(4)=7$ are the only primes not decomposable. Except for $p(1)=2$, $p(2)=3$ and $p(4)=7$, the decomposition into weight × level + jump of prime numbers is:

• A000040(n) = A117078(n) × A117563(n) + A001223(n) (l(n) = A118534(n)).

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 $p(n)$ different from 2, 3 and 7, we have:

• $l(n)=p(n)-g(n)=2\times p(n)-p(n+1)=k(n)\times L(n),\,$

• $p(n)=l(n)+g(n)=k(n)\times L(n)+g(n),\,$

• $\gcd(g(n),2)=2,\,$

• $\gcd(p(n),g(n))=\gcd(p(n)-g(n),g(n))=\gcd(l(n),g(n))=\gcd(L(n),g(n))=\gcd(k(n),g(n))=1,\,$

• $3\leq k(n)\leq l(n),\,$

• $1\leq L(n)\leq l(n)/3,\,$

• $2\leq g(n)\leq k(n)-1,\,$

• $2\times g(n)+1\leq p(n).\,$

Primes classified by weight

For primes classified by weight (Cf. A162175) (primes for which $k(n)\leq L(n)\,$), we have:

• $g(n)+1\leq k(n)\leq {\sqrt {l(n)}}\leq L(n)\leq {\frac {l(n)}{3}}.\,$

82,89 % of primes $p(n)$ are classified by weight for $n\leq 5\cdot 10^{7}$.

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 $k(n)>L(n)$), we have:

• $L(n)<{\sqrt {l(n)}}<k(n)\leq l(n),\,$

• $L(n)+2\leq g(n)+1\leq k(n)\leq l(n).\,$

17,11 % of primes $p(n)$ are classified by level for $n\leq 5\cdot 10^{7}$.

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 $p(n)$ is a lesser of twin prime greater than $3$ then $p(n)$ has a weight of $3$. If $p(n)$ has a weight of $3$ then $p(n)$ is a lesser of twin prime greater than $3$.

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 $k\geq 3\,$ which is not a multiple of 2;

• Conjecture 3: The number of primes of level $L$ is infinite for any $L\geq 1\,$ 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.

 

• Conjecture 7: If the jump g(n) is not a multiple of 6 then l(n) is a multiple of 3. (trivial)

• Conjecture 8: If l(n) is not a multiple of 3 then jump the g(n) is a multiple of 6. (trivial)

  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:

• A005408(n) = A090368(n-1) × A184726(n) + 2 (l(n) = A005408(n-1)).

Plot of log(A090368) vs log(A184726) for n ≤ 10^4 (OEIS graph):

Decomposition of even numbers

If the decomposition is possible, we have:

• A005843(n) = A090369(n-1) × A184727(n) + 2 (l(n) = A005843(n-1)).

Plot of log(A090369) vs log(A184727) for n ≤ 10^4 (OEIS graph):

Decomposition of composite numbers

If the decomposition is possible, we have:

• A002808(n) = A130882(n) × A179621(n) + A073783(n) (l(n) = A179620(n)).

Plot of log(A130882) vs log(A179621) for n ≤ 10^4 (OEIS graph):

Decomposition of semiprimes

If the decomposition is possible, we have:

• A001358(n) = A130533(n) × A184729(n) + A065516(n) (l(n) = A184728(n)).

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:

• A014612(n) = A130650(n) × A184753(n) + A114403(n) (l(n) = A184752(n)).

Plot of log(A130650) vs log(A184753) for n ≤ 10^4 (OEIS graph):

Decomposition of lucky numbers

If the decomposition is possible, we have:

• A000959(n) = A130889(n) × A184828(n) + A031883(n) (l(n) = A184827(n)).

Plot of log(A130889) vs log(A184828) for n ≤ 10^4 (OEIS graph):

Decomposition of prime powers

If the decomposition is possible, we have:

• A000961(n) = A184829(n) × A184831(n) + A057820(n) (l(n) = A184830(n)).

Plot of log(A184829) vs log(A184831) for n ≤ 10^4 (OEIS graph):

Decomposition of squarefree numbers

If the decomposition is possible, we have:

• A005117(n) = A184832(n) × A184834(n) + A076259(n) (l(n) = A184833(n)).

Plot of log(A184832) vs log(A184834) for n ≤ 10^4 (OEIS graph):

Decomposition of triangular numbers

If the decomposition is possible, we have:

• A000217(n) = A130703(n) × A184219(n) + (n + 1) ((l(n) = A184218(n) = A000096(n-2)).

Plot of log(A130703) vs log(A184219) for n ≤ 10^4 (OEIS graph):

Decomposition of squares

If the decomposition is possible, we have:

• A000290(n) = A133150(n) × A184221(n) + A005408(n) (l(n) = A184220(n) = A008865(n-1)).

Plot of log(A133150) vs log(A184221) for n ≤ 10^4 (OEIS graph):

Decomposition of pentagonal numbers

If the decomposition is possible, we have:

• A000326(n) = A133151(n) × A184751(n) + A016777(n) (l(n) = A184750(n)).

Plot of log(A133151) vs log(A184751) for n ≤ 10^3 (OEIS graph):

Sequences

Sequences related to the decomposition.

See also

• Classifications of prime numbers

External links

• arXiv:0711.0865 [math.NT]: Decomposition into weight * level + jump and application to a new classification of primes, 2007 - 2010, Rémi Eismann.