Rémi Eismann<p><a href="https://mathstodon.xyz/tags/decompwlj" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>decompwlj</span></a> ➡️ It's a decomposition of positive integers. The weight is the smallest 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. So to decompose a(n), we need a(n+1) with a(n+1)>a(n) (strictly increasing sequence), the decomposition is possible if a(n+1)<3/2×a(n) and we have the unique decomposition a(n) = weight × level + jump.</p><p>We see the fundamental theorem of arithmetic and the sieve of Eratosthenes in the decomposition into weight × level + jump of natural numbers. For natural numbers, the weight is the smallest prime factor of (n-1) and the level is the largest proper divisor of (n-1). Natural numbers classified by level are the (primes + 1) and natural numbers classified by weight are the (composites +1).</p><p>For prime numbers, this decomposition led to a new classification of primes. Primes classified by weight follow Legendre conjecture and i conjecture that primes classified by level rarefy. I think this conjecture is very important for the distribution of primes.</p><p>It's easy to see and prove that lesser of twin primes (>3) have a weight of 3. So the twin primes conjecture can be rewritten: there are infinitely many primes that have a weight of 3.</p><p>I am not mathematician so i decompose sequences to promote my vision of numbers. By doing these decompositions, i apply a kind of sieve on each sequences.</p><p>➡️ <a href="https://oeis.org/wiki/Decomposition_into_weight_*_level_%2B_jump" rel="nofollow noopener noreferrer" translate="no" target="_blank"><span class="invisible">https://</span><span class="ellipsis">oeis.org/wiki/Decomposition_in</span><span class="invisible">to_weight_*_level_%2B_jump</span></a></p><p><a href="https://mathstodon.xyz/tags/math" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>math</span></a> <a href="https://mathstodon.xyz/tags/maths" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>maths</span></a> <a href="https://mathstodon.xyz/tags/mathematics" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>mathematics</span></a> <a href="https://mathstodon.xyz/tags/sequences" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>sequences</span></a> <a href="https://mathstodon.xyz/tags/OEIS" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>OEIS</span></a> <a href="https://mathstodon.xyz/tags/NumberTheory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>NumberTheory</span></a> <a href="https://mathstodon.xyz/tags/PrimeNumbers" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>PrimeNumbers</span></a> <a href="https://mathstodon.xyz/tags/JavaScript" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>JavaScript</span></a> <a href="https://mathstodon.xyz/tags/php" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>php</span></a> <a href="https://mathstodon.xyz/tags/graph" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>graph</span></a> <a href="https://mathstodon.xyz/tags/3D" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>3D</span></a> <a href="https://mathstodon.xyz/tags/classification" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>classification</span></a> <a href="https://mathstodon.xyz/tags/primes" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>primes</span></a> <a href="https://mathstodon.xyz/tags/threejs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>threejs</span></a> <a href="https://mathstodon.xyz/tags/webGL" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>webGL</span></a> <a href="https://mathstodon.xyz/tags/integer" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>integer</span></a> <a href="https://mathstodon.xyz/tags/decomposition" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>decomposition</span></a> <a href="https://mathstodon.xyz/tags/arithmetic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>arithmetic</span></a> <a href="https://mathstodon.xyz/tags/numbers" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>numbers</span></a> <a href="https://mathstodon.xyz/tags/theory" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>theory</span></a> <a href="https://mathstodon.xyz/tags/equation" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>equation</span></a> <a href="https://mathstodon.xyz/tags/graphs" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>graphs</span></a> <a href="https://mathstodon.xyz/tags/sieve" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>sieve</span></a> <a href="https://mathstodon.xyz/tags/fundamental" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>fundamental</span></a> <a href="https://mathstodon.xyz/tags/theorem" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>theorem</span></a> <a href="https://mathstodon.xyz/tags/arithmetic" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#<span>arithmetic</span></a></p>