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How many positive integers less than 2000 are of the form \(x^n\) for some positive integer \(x\) and \(n\ge 2?\)

Oct 12, 2019

#1
+1

a=2; b=2;c= a^b; if(c<=2000, goto4, goto5);printc,", ",; a++;if(a<45, goto2, 0);a=2;b++;if(b<45, goto2,0)

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  400  441  484  512  529  576  625  676  729  784  841  900  961  1000  1024  1089  1156  1225  1296  1331  1369  1444  1521  1600  1681  1728  1764  1849  1936  >>Total =  58

Oct 12, 2019
#2
+7

Squares: there are 44 of them (44*44=1936 but 45*45=2025)

Cubes: there are 12 of them (up to 12*12*12=1728) but exclude 1^3, 4^3, and 9^3 as those are also squares, so just 9 unique ones.

Fourth powers (and all non-prime powers)—are included under lower-ranked exponents

Fifth powers: 4 of them (up to 4*4*4*4*4=1024, but exclude 1^5 and 4^5 as they are also squares, so just 2 unique ones

Seventh powers: Just 1 unique one: 2^7=128.

Once we get up to 11th powers or higher, the lowest unique example is 2^11=2048 which is too big.

So in total, there are 44+9+2+1=56 such numbers up to 2000.

Oct 13, 2019
#3
+1

Thank you, SVS!

CalculatorUser  Oct 13, 2019
#4
+1

Yes thanks guys and girls Oct 13, 2019
#5
+5

No Problem, Thanks for the graph Melody!

SVS2652  Oct 13, 2019