Decomposition into weight × level + jump

An extension of the fundamental theorem of arithmetic and a new way to see the numbers: the decomposition into weight × level + jump.

[decompwlj]

Decomposition into weight × level + jump with 3D graphs (WebGL three.js), 2D graphs and first 500 terms. This decomposition is a generalization of the sieve of Eratosthenes. 1000 sequences decomposed (natural numbers, prime numbers, triangular numbers, composites...). Rémi Eismann

1000 sequences decomposed into weight × level + jump on my website. Before continuing, I would like to have an independent verification of my data. There is a complete dump of my database on the home page (.sql.zip, ~105 Mo, central table “sequences” and 1 table per sequence).

[decompwlj]

New decompositions:

A145749: Numbers n such that sigma(n)+phi(n)=sigma(n+1)+phi(n+1).

A152470: Largest of three consecutive primes whose sum is a prime.

A156849: Numbers n such that n^2 == 2 mod (23^2).

A157468: Primes of the form sqrt(p-1)-1, where p is a prime.

Now 1000 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A141165: Primes of the form 9*x^2+7*x*y-5*y^2.

A157483: Numbers n such that n-1 and n+1 are divisible by exactly 3 primes, counted with multiplicity.

A158714: Primes p such that p1 = ceil(p/2) + p is prime and p2 = floor(p1/2) + p1 is prime.

A158913: Primes p such that there is a composite c with sigma(p) = sigma(c).

Now 996 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A127341: Primes that can be written as the sum of 13 consecutive primes.

A136333: Numbers containing only digits coprime to 10 in their decimal representation.

A137812: Left- or right-truncatable primes.

A141161: Primes of the form 4*x^2+6*x*y-7*y^2.

Now 992 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A118954: Numbers that cannot be written as 2^k + prime.

A119449: Primes with even digit sum.

A121943: Numbers k such that the central binomial coefficient C(2k,k) is divisible by k^2.

A125022: Even numbers with a unique partition as the sum of 2 squares x^2+y^2, divided by 2.

Now 988 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A107181: Primes of the form 8x^2 + 9y^2.

A111046: Difference between squares of twin prime pairs.

A117048: Prime numbers that are expressible as the sum of two positive triangular numbers.

A118134: Primes p such that 4p is the sum of two consecutive primes.

Now 984 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A101095: Fourth difference of fifth powers (A000584).

A103664: Primes p such that the number of divisors of p-1 is less than the number of divisors of p+1.

A105441: Numbers with at least two odd prime factors (not necessarily distinct).

A106881: Primes of the form x^2+xy+9y^2.

Now 980 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A097103: Numbers n that are the hypotenuse of exactly 22 distinct integer sided right triangles, i.e. n^2 can be written as a sum of two squares in 22 ways.

A097933: Primes such that p divides 3^((p-1)/2) - 1.

A099204: A variation on Flavius's sieve (A000960): Start with the natural numbers; at the k-th sieving step, remove every p-th term of the sequence remaining after the (k-1)-st sieving step, where p is the k-th prime; iterate.

A100493: a(n) = n + n-th semiprime.

Now 976 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A090693: Positive numbers n such that n^2 - 2n + 2 is a prime.

A091300: Nonprimes of form 6k+1.

A092022: Numbers n such that 16n + 3 is prime.

A092192: Semiprimes that are the sum of two successive semiprimes.

Now 972 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A087054: Primes of the form pq + qr + rp where p, q and r are distinct primes.

A087248: Squarefree abundant numbers.

A089194: Primes p such that p-1 and p+1 are cube- or higher power-free.

A089589: Iban numbers (the letter i is banned from the English name of the number).

Now 968 sequences decomposed into weight × level + jump on my website (https + responsive).

#math #maths #mathematics #numbers #sequences #sequence #prime numbers #OEIS

[decompwlj]

New decompositions:

A080218: Monotonically increasing sequence such that every positive integer n appears if and only if d(n) doesn't (d(n)=number of divisors of n, A000005).

A081311: Numbers that can be written as sum of a prime and an 3-smooth number.

A081605: Numbers having at least one 0 in their ternary representation.

A086381: Numbers n such that p=n^2+2 and p+2 are primes.

Now 964 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A074627: Numbers n such that sigma(n) is divisible by 6.

A076056: Primes which when read backwards are composite numbers.

A077064: Squarefree numbers of form prime - 1.

A079545: Primes of the form x^2 + y^2 + 1 with x,y >= 0.

Now 960 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A068501: Values m such that the consecutive pair parameters(m,m+1) generate Pythagorean triples whose odd terms are both prime.

A069346: Primes of the form n - Omega(n), where Omega(n) is the number of prime factors of n, A001222(n).

A071403: Which squarefree number is prime? a(n)-th squarefree number equals n-th prime.

A072225: Numbers n such that prime(n) + prime(n+1) + prime(n+2) is prime.

Now 956 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A061384: Numbers n with property that sum of digits = number of digits.

A061426: Geometric mean of the digits = 2. In other words the product of the digits is = 2^k where k is the number of digits.

A062115: Numbers with no prime substring in their decimal expansion.

A066825: a(1) = 1; set of digits of a(n)^2 is a subset of the set of digits of a(n+1)^2.

Now 952 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A054402: Numbers that are the sum of a positive square and a positive cube in more than one way.

A055638: Numbers n for which sigma(n^2) is prime.

A059269: Numbers n for which the number of divisors, tau(n), is divisible by 3.

A059708: Numbers n such that all digits have same parity.

Now 948 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A052020: Sum of digits of n is a prime proper factor of n.

A052044: Numbers n such that n^3 lacks the digit zero in its decimal expansion.

A052485: Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.

A053868: Numbers n such that sum of divisors of n less than n is odd.

Now 944 sequences decomposed into weight × level + jump on my website (https + responsive).

[decompwlj]

New decompositions:

A045746: Numbers n such that the sum of the divisors of n is a triangular number.

A045942: Numbers n such that the factorizations of n through n+5 have the same number of primes (including multiplicities).

A048521: Primes expressible as the sum of an integer plus its digit sum.

A050931: Solutions c of cot(2*Pi/3)*(-(a+b+c)*(-a+b+c)*(-a+b-c)*(a+b-c))^(1/2)=a^2+b^2-c^2, c>a,b integers.

Now 940 sequences decomposed into weight × level + jump on my website (https + responsive).