Editing Powerful number (section)

{{Short description|Numbers whose prime factors all divide the number more than once}}
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|text=<math>\begin{align}144000&=2^7\times 3^2\times 5^3\\ &=2^3\times 2^4\times 3^2 \times 5^3\\ &=(2\times5)^3 \times (2^2\times 3)^2\end{align} </math>

<small>144000 is a powerful number. <br>Every exponent in its [[prime factorization]] is larger than 1. <br>It is the product of a square and a cube.</small>
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A '''powerful number''' is a [[positive integer]] ''m'' such that for every [[prime number]] ''p'' dividing ''m'', ''p''<sup>2</sup> also divides ''m''. Equivalently, a powerful number is the product of a [[Square number|square]] and a [[Cube (arithmetic)|cube]], that is, a number ''m'' of the form ''m'' = ''a''<sup>2</sup>''b''<sup>3</sup>, where ''a'' and ''b'' are positive integers. Powerful numbers are also known as '''squareful''', '''square-full''', or '''2-full'''. [[Paul Erdős]] and [[George Szekeres]] studied such numbers and [[Solomon W. Golomb]] named such numbers ''powerful''.

The following is a list of all powerful numbers between 1 and 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, ... {{OEIS|id=A001694}}.
[[File:Powerful_numbers_up_to_100.svg|thumb|upright=0.5|Powerful numbers up to 100 with prime factors colour-coded &ndash; 1 is a special case]]