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  • Definitions - prime numbers

    Let be p(n) the sequence of prime numbers (A000040 on the OEIS), we define :

    the jump of p(n) by (A001223)
    g(n) = p(n+1) - p(n) ;

    l(n) (A118534) by
    l(n) = largest l such that g(n) = p(n) mod l, 0 if no such l exists, or
    l(n) = p(n) - g(n) if p(n) - g(n) > g(n), 0 otherwise ;

    the weight (A117078) by
    k(n) = smallest k such that g(n) = p(n) mod k, 0 if no such k exists, or
    k(n) = smallest k greater than g(n), that divides l(n), 0 if l(n)=0 ;

    The level by (A117563)
    L(n) = l(n) / k(n), 0 if k(n) = 0.

    The weight k(n) is the smallest such that in the Euclidean division of p(n) by its weight k(n), the quotient is the level L(n), and the remainder is the jump g(n).
    We have p(n) = k(n) * L(n) + g(n) (A000040(n) = A117078(n) * A117563(n) + A001223(n)) when l(n) is different from 0 which can be rewritten when p(n+1) < (3/2) * p(n).

    Principles of classification : if for p(n), l(n) = k(n) = L(n) = 0 then p(n) is not classified. If for p(n), k(n) > L(n) then p(n) is classified by level if not p(n) is classified by weight.