Potent number

A potent number is a natural number with the property that for every prime divisor of is also a divisor of . Equivalent to this, a potent number is the product of a square number and a cube number: with natural numbers and . Paul Erdős and George Szekeres investigated such figures, Solomon W. Golomb called them powerful . ${\displaystyle m}$ ${\displaystyle p}$ ${\displaystyle m}$ ${\displaystyle p^{2}}$ ${\displaystyle m}$ ${\displaystyle m=a^{2}b^{3}}$ ${\displaystyle a}$ ${\displaystyle b}$

List of all potent numbers from 1 to 1000:

1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000.

Follow A001694 in On-Line Encyclopedia of Integer Sequences

The series over the reciprocal values ​​of all potent numbers can be represented in a closed manner with the help of the Riemann ζ function . (Golomb, 1970)

${\displaystyle \sum _{n\,{\text{potent}}}{\frac {1}{n}}=\prod _{p}\left(1+{\frac {1}{p\,(p-1)}}\right)={\frac {\zeta (2)\,\zeta (3)}{\zeta (6)}}={\frac {315}{2\pi ^{4}}}\zeta (3)=1{,}9435964368207592050570...}$

Here is the Apéry constant , for which there is no exact representation as there is for straight arguments of the Riemann zeta function . Their numerical value is . ${\displaystyle \zeta (3)}$ ${\displaystyle \zeta (3)=1{,}20205690315959428539...}$

Web links

• The abc conjecture