User:Daniel Forgues/Workspace

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[Power]free, [power]ful ([power]some, not [power]free) and Golomb's [power]-full or [power]full or n-full numbers

Squarefree, squareful (squaresome, not squarefree) and Golomb's square-full or squarefull or 2-full numbers

Squarefree

Weisstein, Eric W., Squarefree, from MathWorld—A Wolfram Web Resource.

A005117 Squarefree numbers: numbers that are not divisible by a square greater than 1.

Characteristic function

A008966 1 if n is squarefree, else 0.

Squarefree kernel of a number

A007947 Largest squarefree number dividing n (the squarefree kernel of n).

A007913 Squarefree part of n: a(n) = smallest positive number m such that n/m is a square.

Asymptotic density

${\frac {1}{\zeta (2)}}\,$

Squareful (squaresome, not squarefree)

Weisstein, Eric W., Squareful, from MathWorld—A Wolfram Web Resource.

A013929 Numbers that are not squarefree. Numbers that are divisible by a square greater than 1. The complement of A005117.

Golomb's square-full or squarefull or 2-full or powerful numbers

Weisstein, Eric W., PowerfulNumber, from MathWorld—A Wolfram Web Resource.

A001694 Powerful numbers, definition (1): if a prime p divides n then p^2 must also divide n (also called squarefull, square-full or 2-full numbers).

Consecutive powerful numbers

A060355 Numbers n such that n and n+1 are a pair of consecutive powerful numbers.

A118893 Numbers n such that n-1 and n are a pair of consecutive powerful numbers.

Number of powerful numbers

A118896 Number of powerful numbers <= 10^n.

Sum of reciprocals

Golomb (1970) showed that the sum over the reciprocals of the powerful numbers is

$\sum _{k=1}^{\infty }{\frac {1}{P_{k}}}={\frac {\zeta (2)\zeta (3)}{\zeta (6)}},\,$ (decimal expansion: A082695)

where $\scriptstyle P_{k}\,$ is the $\scriptstyle k\,$^th 2-full number (powerful numbers being of the form $\scriptstyle a^{2}b^{3},\ a,\,b\,\geq 1\,$.

Cubefree, cubeful (cubesome, not cubefree) and Golomb's cube-full or cubefull or 3-full numbers

Cubefree

Weisstein, Eric W., Cubefree, from MathWorld—A Wolfram Web Resource.

A004709 Cube-free numbers: numbers that are not divisible by any cube > 1.

Asymptotic density

${\frac {1}{\zeta (3)}}\,$

Cubeful (cubesome, not cubefree)

Weisstein, Eric W., Cubeful, from MathWorld—A Wolfram Web Resource.

A046099 Numbers that are not cube-free. Numbers divisible by a cube greater than 1. Complement of A004709.

Golomb's cube-full or cubefull or 3-full numbers

Weisstein, Eric W., Cubefull, from MathWorld—A Wolfram Web Resource.

A036966 3-full (or cube-full, or cubefull) numbers: if a prime p divides n then so does p^3.

Biquadratefree, biquadrateful (biquadratesome, not biquadratefree) and Golomb's biquadrate-full or biquadratefull or 4-full numbers

Biquadratefree

Weisstein, Eric W., Biquadratefree, from MathWorld—A Wolfram Web Resource.

A046100 Biquadratefree numbers.

Asymptotic density

${\frac {1}{\zeta (4)}}\,$

Biquadrateful (biquadratesome, not biquadratefree)

Weisstein, Eric W., Biquadrateful, from MathWorld—A Wolfram Web Resource.

A046101 Biquadrateful numbers.

Golomb's biquadrate-full or biquadratefull or 4-full numbers

Weisstein, Eric W., Biquadratefull, from MathWorld—A Wolfram Web Resource.

A036967 4-full numbers: if a prime p divides n then so does p^4.

Achilles number

An Achilles number (Cf. A052486) is a positive integer that is imperfectly powerful, i.e. it is powerful but imperfect (not a perfect power).

Thus there are two kinds of powerful numbers:

Perfectly powerful numbers (perfect powers greater than 1) (A072777 Powers of squarefree numbers which are not squarefree.)

Imperfectly powerful numbers (Achilles numbers) A052486

Sequences: floor (1/2 + r n mod 1), r ∈ ℝ , r > 1, n   ≥   1

Do we have a 1-to-1 correspondence between any real number

r > 1

and binary spectrums from

floor (1/2 + r  n mod 1), r ∈ ℝ , r > 1, n   ≥   1

? If so, can we get back the real number from its spectrum?
A??????

floor (1/2 + π n mod 1), n   ≥   1.

{0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, ...}

A??????

floor (1/2 + ϕ n mod 1), n   ≥   1.

{, ...}

A??????

floor (1/2 + (22 / 7 ) n mod 1), n   ≥   1.

{0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 1, ...}

References

• Golomb, S. W., Powerful Numbers, Amer. Math. Monthly 77, pp. 848–855, 1970.