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powerful numbers
An integer  $n$  is called powerful if, for every prime  $p$  dividing  $n$,  $p^2$  also divides  $n$.

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  $a^2b^3$, for  $a,b \ge 1$.

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  $\zeta(2)\zeta(3)/\zeta(6)\approx 1.9435964\dots$.

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

\[ \frac{\zeta(3/2)}{\zeta(3)}\sqrt{n}+\frac{\zeta(2/3)}{\zeta(2)}\sqrt[3]{n}+o(\sqrt[6]{n})\,. \]

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

Pictorial representation of remainders (mod 2, 3, ...,11) frequency. For a table of values and more details click here
remainders of powerful numbers

A graph displaying how many powerful numbers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of powerful numbers
Useful links Mathworld, Powerful Number
Wikipedia, Powerful number
OEIS, Sequence A001694
Powerful numbers can also be... (you may click on names or numbers and on + to get more values)

ABA 32 64 72 + 9999982312712 aban 16 25 27 + 1000000000000 abundant 36 72 100 + 50000000 Achilles 72 108 200 + 50000000000000 admirable 1372 476656 570375 alternating 16 25 27 + 987656329 amenable 16 25 32 + 1000000000 apocalyptic 243 361 529 + 29929 arithmetic 27 49 125 + 9948825 astonishing 27 216 automorphic 25 625 8212890625 binomial 36 1225 19600 + 1882672131025 brilliant 25 49 121 + 999002449 c.decagonal 361 116281 37442161 + 3882078149401 c.heptagonal 841 1331 512072 + 546848356574521 c.nonagonal 1225 1413721 1631432881 1882672131025 c.octagonal 25 49 81 + 999999898700625 c.pentagonal 16 1156 22801 + 68366835443776 c.square 25 841 28561 + 43892069261881 c.triangular 64 361 6241 + 319312633001041 cake 64 576 2048 + 75203584 Canada 125 compositorial 1728 5267275776000 congruent 125 216 343 + 9984600 constructible 16 32 64 + 562949953421312 cube 27 64 125 + 999970000299999 Cullen 25 Cunningham 288 675 9800 + 511643454094368 Curzon 81 125 441 + 198218241 d-powerful 1323 2048 4225 + 9972964 de Polignac 40401 62001 96721 + 99620361 decagonal 27 4000 469567 + 38637897507 deceptive 237169 deficient 16 25 27 + 9991921 dig.balanced 49 108 169 + 199967881 droll 72 800 5184 + 994721136640000 Duffinian 16 25 27 + 9999392 eban 32 36 64 + 64032004000000 economical 16 25 27 + 20000000 emirpimes 49 169 289 + 99460729 enlightened 256 2048 2304 + 373714754427 equidigital 16 25 27 + 19998784 eRAP 265225 616225 13213225 + 519926081481 esthetic 32 121 343 + 123456787654321 evil 27 36 72 + 999824400 Fibonacci 144 Friedman 25 121 125 + 995328 frugal 125 128 243 + 999887641 gapful 100 108 121 + 100000000000 Gilda 49 happy 32 49 100 + 10000000 Harshad 27 36 72 + 10000000000 heptagonal 81 5929 2307361 + 350709705290025 hex 169 32761 6355441 + 46399815451081 hexagonal 1225 1413721 74024028 + 194838725161125 highly composite 36 hoax 361 1600 2401 + 95520600 Hogben 343 7906143973 house 32 1175056 hungry 144 iban 27 72 100 + 774400 idoneal 16 25 72 impolite 16 32 64 + 562949953421312 inconsummate 216 432 441 + 996872 interprime 64 72 81 + 99904500 Jordan-Polya 16 32 36 + 995515121664000 junction 216 1024 2025 + 99880036 Kaprekar 5292 katadrome 32 64 72 + 972 Leyland 32 100 512 + 17832200896512 lucky 25 49 169 + 9948123 Lynch-Bell 36 128 216 + 9176328 magnanimous 16 25 32 + 22801 metadrome 16 25 27 + 134689 modest 49 82369 312481 + 1976030303 Moran 27 nialpdrome 32 64 72 + 999887641000000 nonagonal 1089 8281 28125 + 708304623404049 nude 36 128 144 + 498389976 O'Halloran 36 oban 16 25 27 + 968 octagonal 225 43681 78408 + 61866420601441 odious 16 25 32 + 1000000000 palindromic 121 343 484 + 900075181570009 pancake 16 121 529 + 937385441796001 panconsummate 36 72 81 + 361 pandigital 108 216 225 + 9876523104 pentagonal 9801 16008300 94109401 + 903638458801 pernicious 25 36 49 + 9999392 persistent 1324890675 1532487609 1854207963 + 98175243600 plaindrome 16 25 27 + 277777788888889 Poulet 1194649 12327121 power 16 25 27 + 49999988518489 practical 16 32 36 + 10000000 prim.abundant 196 15376 342225 + 99198099 pronic 72 83232 456300 + 127910863523592 Proth 25 49 81 + 274878955521 pseudoperfect 36 72 100 + 1000000 repunit 121 343 400 7906143973 Rhonda 1568 5832 15625 + 21353351168 Ruth-Aaron 16 25 49 + 996383276100 Saint-Exupery 202500 1620000 5467500 + 998557832475000 self 64 108 121 + 999887641 semiprime 25 49 121 + 99460729 sliding 25 200 2000 + 250000000000000 Smith 27 121 576 + 99630728 square 16 25 36 + 999999961946176 star 121 11881 226981 + 107341182792481 straight-line 432 864 3456 strobogrammatic 6889 69169 109181601 super Niven 36 100 200 + 45000000000 super-d 81 169 784 + 9984600 superabundant 36 tau 36 72 108 + 1000000000 taxicab 28149336 225194688 760032072 + 993321157865472 tetrahedral 19600 tetranacci 108 triangular 36 1225 29403 + 194838725161125 tribonacci 81 3136 10609 trimorphic 25 49 125 + 8212890625 uban 16 25 27 + 99099040000000 Ulam 16 36 72 + 9980928 undulating 121 343 484 + 69696 unprimeable 200 324 512 + 10000000 untouchable 216 288 324 + 990584 upside-down 64 8192 22128988 + 22817137939288 vampire 2187 186624 10396800 + 5894137611 wasteful 36 72 100 + 9999392 Zuckerman 36 128 144 + 9916826112 Zumkeller 108 216 432 + 100000 zygodrome 7744 665500 774400 + 992255544007744