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%I #13 Jun 07 2016 11:29:32
%S 13140625,36859543552,49762009476,87169610025,3324163986441,
%T 2988330556640625,10155995666841600,28920784535654400,
%U 34328125000000000,65388757868609536,101445409544601600,275625000000000000,428123439576907776
%N Perfect powers m^k such that m^k = a^2 + b^4 = c^3 + d^5 for some positive integers a, b, c, d.
%C Intersection of A001597, A100293, A111925.
%C 3625^2 = 13140625 is the least number with this property.
%C Sequence is infinite because if m^k = a^2 + b^4 = c^3 + d^5 is a term, then (m*t^60)^k = (a*t^(30*k))^2 + (b*t^(15*k))^4 = (c*t^(10*k))^3 + (d*t^(12*k))^5 is also a term for every t>1. - _Giovanni Resta_, Jun 07 2016
%e 13140625 is a term because 13140625 = 3625^2 = 2625^2 + 50^4 = 150^3 + 25^5.
%o (PARI) isA111925(n)=for(b=1,sqrtnint(n-1,4), if(issquare(n-b^4), return(1))); 0
%o isA100293(n)=for(y=1, sqrtnint(n-1, 5), if(ispower(n-y^5, 3), return(1))); 0
%o list(lim)=my(v=List(), b4, t); for(e=2,logint(lim\=1,2), for(m=2,sqrtnint(lim,e), t=m^e; if(isA111925(t) && isA100293(t), listput(v, t)))); Set(v) \\ _Charles R Greathouse IV_, Jun 07 2016
%Y Cf. A001597, A100293, A111925.
%K nonn
%O 1,1
%A _Altug Alkan_, Jun 07 2016
%E a(2)-a(13) from _Giovanni Resta_, Jun 07 2016
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