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  • Definitions - natural numbers

    Let be n the sequence of natural numbers (A000027 on the OEIS), we define :

    the jump of n by (A000012)
    d(n) = 1 ;

    l(n) by
    l(n) = largest l such that 1 = n mod l, 0 if no such l exists, or
    l(n) = n - 1 if n - 1 > 1, 0 otherwise ;

    the weight by (A020639)
    k(n) = smallest k such that 1 = n mod k, 0 if no such k exists, or
    k(n) = smallest k greater than 1, that divides n - 1 if n > 2, 0 otherwise ;

    the level by (A032742)
    L(n) = (n - 1) / k(n) if n > 2, 0 otherwise.

    The weight is the smallest prime factor of n - 1 and the level is the largest proper divisor of n - 1.
    We have n = k(n) * L(n) + 1 (A000027(n) = A020639(n-1) * A032742(n-1) + 1) when n > 2.

    Principles of classification : For n <= 2, n is not classified. If for n, k(n) > L(n) then n is classified by level if not n is classified by weight.