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perfect powers
An integer  $n$  is a perfect power if it is equal to  $m^k$, for  $m0$">  and  $k\ge2$.

The sum of the reciprocals of the perfect powers larger than 1, including duplicates, like  $2^6 = 4^3$, is equal to 1, exactly like the sum of the reciprocals of  $p-1$, where  $p1$">  ranges among the perfect powers, but without duplicates, i.e.,

\[ \sum_{k=2}^{\infty}\sum_{n=2}^{\infty}\frac{1}{n^k}=\sum_{p}\frac{1}{p-1}=1\,,\]

The first perfect powers are 1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 81, 100, 121, 125, 128, 144, 169, 196, 216, 225, 243, 256, 289, 324, 343, 361 more terms

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

A graph displaying how many perfect powers are multiples of the primes p from 2 to 71. In black the ideal line 1/p.
divisibility of perfect powers
Useful links Mathworld, PerfectPower
Wikipedia, Perfect power
OEIS, Sequence A001597
Perfect powers can also be... (you may click on names or numbers and on + to get more values)

ABA 32 64 81 + 9986219369604 aban 16 25 27 + 1000000000000 abundant 36 100 144 + 49984900 alternating 16 25 27 + 987656329 amenable 16 25 32 + 1000000000 apocalyptic 243 361 529 + 29929 arithmetic 27 49 125 + 9938375 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 755161 + 608963290321 c.nonagonal 1225 1413721 1631432881 1882672131025 c.octagonal 25 49 81 + 49999988518489 c.pentagonal 16 1156 22801 + 3465632747641 c.square 25 841 28561 + 43892069261881 c.triangular 64 361 6241 + 3258631508224 cake 64 576 2048 + 75203584 Canada 125 compositorial 1728 congruent 125 216 343 + 9938375 constructible 16 32 64 + 35184372088832 cube 27 64 125 + 49998717504000 Cullen 25 Curzon 81 125 441 + 198218241 d-powerful 2048 4225 4624 + 9972964 de Polignac 40401 62001 96721 + 99620361 decagonal 27 deceptive 237169 deficient 16 25 27 + 9991921 dig.balanced 49 169 216 + 199967881 droll 5184 57600 373248 + 46902116614144 Duffinian 16 25 27 + 9991921 eban 32 36 64 + 36000000000000 economical 16 25 27 + 19998784 emirpimes 49 169 289 + 99460729 enlightened 256 2048 2304 + 373714754427 equidigital 16 25 27 + 19998784 eRAP 265225 616225 13213225 + 519926081481 esthetic 32 121 343 + 1234567654321 evil 27 36 125 + 999824400 Fibonacci 144 Friedman 25 121 125 + 992016 frugal 125 128 243 + 999887641 gapful 100 121 225 + 100000000000 Gilda 49 happy 32 49 100 + 10000000 Harshad 27 36 81 + 10000000000 heptagonal 81 5929 2307361 + 4797839017609 hex 169 32761 6355441 + 46399815451081 hexagonal 1225 1413721 1631432881 1882672131025 highly composite 36 hoax 361 1600 2401 + 91011600 Hogben 343 house 32 1175056 hungry 144 iban 27 100 121 + 774400 idoneal 16 25 impolite 16 32 64 + 35184372088832 inconsummate 216 441 1521 + 962361 interprime 64 81 144 + 99880036 Jordan-Polya 16 32 36 + 47775744000000 junction 216 1024 2025 + 99880036 katadrome 32 64 81 + 961 Leyland 32 100 512 33554432 lucky 25 49 169 + 9941409 Lynch-Bell 36 128 216 + 1382976 magnanimous 16 25 32 + 22801 metadrome 16 25 27 + 134689 modest 49 82369 312481 + 1666027489 Moran 27 nialpdrome 32 64 81 + 10000000000000 nonagonal 1089 8281 978121 + 5996832038649 nude 36 128 144 + 494439696 O'Halloran 36 oban 16 25 27 + 900 octagonal 225 43681 8473921 + 318907548961 odious 16 25 32 + 1000000000 palindromic 121 343 484 + 9420645460249 pancake 16 121 529 + 27594051024016 panconsummate 36 81 121 + 361 pandigital 216 225 38025 + 9814072356 pentagonal 9801 94109401 903638458801 pernicious 25 36 49 + 9998244 persistent 1532487609 2081549376 3285697041 + 96438197025 plaindrome 16 25 27 + 44444448888889 Poulet 1194649 12327121 powerful 16 25 27 + 49999988518489 practical 16 32 36 + 10000000 prim.abundant 196 15376 342225 1032256 Proth 25 49 81 + 274878955521 pseudoperfect 36 100 144 + 1000000 repunit 121 343 400 Rhonda 5832 15625 342225 + 5547121441 Ruth-Aaron 16 25 49 + 996383276100 Saint-Exupery 202500 12960000 147622500 + 49438476562500 self 64 121 400 + 999887641 semiprime 25 49 121 + 99460729 sliding 25 2500 250000 + 25000000000000 Smith 27 121 576 + 99281296 square 16 25 36 + 49999988518489 star 121 11881 226981 + 1095432263641 strobogrammatic 6889 69169 109181601 super Niven 36 100 400 + 44100000000 super-d 81 169 784 + 9979281 superabundant 36 tau 36 128 225 + 1000000000 taxicab 6080256576 49991146569 389136420864 + 36443545848801 tetrahedral 19600 triangular 36 1225 41616 + 1882672131025 tribonacci 81 3136 10609 trimorphic 25 49 125 + 8212890625 uban 16 25 27 + 49098049000000 Ulam 16 36 243 + 9796900 undulating 121 343 484 + 69696 unprimeable 324 512 625 + 10000000 untouchable 216 324 576 + 984064 upside-down 64 8192 45346756 + 43211599876 vampire 2187 186624 19847025 + 5267275776 wasteful 36 100 144 + 9998244 Zuckerman 36 128 144 + 6318342144 Zumkeller 216 1000 1728 + 100000 zygodrome 7744 774400 77440000 + 8844449977444