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%I #7 Jan 19 2019 04:15:43
%S 47775744,22143375000,23328000000
%N Numbers z such that z^8 = x^6 + y^7 for some integers x, y >= 1.
%C Also in the sequence: 4001504141376, 4275897935643, 11284439629824, 100192997081088, ... (with b = 63, 51, 72, 96, ... cf. below).
%C Consider a solution (x,y,z) of x^6 + y^7 = z^8. For any m, (x*m^28, y*m^24, z*m^21) is also a solution. Reciprocally, if (x/m^28, y/m^24, z/m^21) is a triple of integers for some m, then this is also a solution. We call primitive a solution for which there is no such m > 1.
%C When S = a^6 + b^14/4 is a square, then z = b^7/2 + sqrt(S) is a solution, with x = a*z and y = b*z. All known solutions are of this form.
%C Sequence A303268 lists the y-values, in the above case equal to b*z where z = a(n), with corresponding x = a*z = (a(n)^8 - A303268(n)^7)^(1/6).
%e a(1) = 47775744 = 2^16*3^6 is in the sequence because for this (2^16*3^6)^8 =: z^8 = (12*z)^7 + (288*z)^6 = (2^18*3^7)^7 + (2^21*3^8)^6.
%e a(2) = 22143375000 and a(3) = 23328000000 are in the sequence because for these z, we have z^8 = (30*z)^7 + x^6 with x = 1350*z resp. x = 1800*z.
%o (PARI) is(z)=for(y=1,sqrtnint(-1+z=z^8,7),ispower(z-y^7,6)&&return(y)) \\ returns the y-value (cf. A303268) if z is in this sequence, 0 else.
%o /* Code below for illustration only, not guaranteed to give a complete list. */
%o S=[]; N=1e5; forstep(b=1,99,1/2, forstep(a=1,N,1/2, issquare(b^14/4+a^6,&r)&& !frac(z=b^7/2+r)&& !print("\n*** "z)&& S=setunion(S,[z])); print1([b])); S
%Y Cf. A300564 (z^4 = x^2 + y^3) and A242183, A300565 (z^5 = x^3 + y^4), A300566 (z^6 = x^4 + y^5), A300567 (z^7 = x^5 + y^6), A302174.
%Y See A303268 for the y-values.
%Y Cf. A303376 (numbers of the form a^6 + b^7).
%K nonn,more,bref,hard
%O 1,1
%A _M. F. Hasler_, May 04 2018
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