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powerful numbers
An integer    is called powerful if, for every prime    dividing  ,    also divides  .

In practice, the set of powerful numbers consists of the number 1 plus all numbers in whose factorizations the primes appears with exponents greater than 1. This set coincides with the set of numbers of the form  , for  .

There are infinite pairs of consecutive powerful numbers, the smallest being (8, 9), but Erdös, Mollin, and Walsh conjectured that there are no three consecutive powerful numbers.

Heath-Brown has shown in 1988 that every sufficiently large natural number is the sum of at most three powerful numbers. Probably the largest number which is not the sum of 3 powerful numbers is 119.

The sum of the reciprocals of the powerful numbers converges to  .

P.T.Bateman & E.Grosswald have proved that the asymptotic number of powerful numbers up to    is given by

The first powerful numbers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200 more terms

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