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%I #3 Mar 30 2012 17:35:02
%S 2,3,498,2266144,272585923
%N Numbers n such that n^7 and a cube are between consecutive squares.
%C No other terms < 10^8. The corresponding sequence for n^5 is A173341. Are there ever more than two perfect powers between consecutive squares?
%C a(6) > 10^10. [From _Donovan Johnson_, Apr 17 2010]
%e 2 is here because 2^7=128 and 5^3=125 are between 11^2=121 and 12^2=144.
%e 3 is here because 3^7=2187 and 13^3=2197 are between 46^2=2116 and 47^2=2209.
%e 498 is here because 498^7 and 1965781^3 are between 2756149047^2 and 2756149048^2.
%e 2266144 is here because 2266144^7 and 674534510965903^3 are between 17518876914709436673663^2 and 17518876914709436673664^2.
%t t={}; Do[n2=Floor[n^(7/2)]; n3=Round[n^(7/3)]; If[n2^2<n3^3<(n2+1)^2 && n2^2<n^7<(n2+1)^2 && n3^3 != n^7, AppendTo[t,n]], {n,10^4}]; t
%Y A097056, A117896
%K nonn
%O 1,1
%A _T. D. Noe_, Feb 16 2010
%E a(5) from _Donovan Johnson_, Apr 17 2010
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