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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

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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