A300567
Numbers z such that z^7 = x^5 + y^6 for some integers x, y >= 1.
Original entry on oeis.org
8192, 7593750, 8605184
Offset: 1
a(1) = 8192 = 2^13 is in the sequence because (2^13)^7 = (2^18)^5 + (2^15)^6, using 18*5 = 15*6 = 90 = 13*7 - 1 and 1 + 1 = 2.
Cf.
A303375 (numbers of the form a^5 + b^6).
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is(z)=for(y=1,sqrtnint(-1+z=z^7,6),ispower(z-y^6,5)&&return(y))
/* Code below for illustration only, not guaranteed to give a complete list. */
S=[]; N=1e5; forstep(b=1,99,1/6, forstep(a=1,N,1/6, issquare(b^12/4+a^5,&r)&& !frac(z=b^6/2+r)&& S=setunion(S,[z])); print1([b])); S
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# See Hayden link. This code is built to identify valid z values based on specific conjectures outlined in the file.
A303268
Least y for which x^6 + y^7 = A300568(n)^8 for some x > 1.
Original entry on oeis.org
573308928, 664301250000, 699840000000
Offset: 1
A300568(1) = 47775744 is the smallest z such that z^8 = x^6 + y^7 for some x, y > 1, and the smallest such y is a(1) = 12*z = 573308928. It then follows that x = (47775744^8 - 573308928^7)^(1/6) = 13759414272 = 288*z.
A300568(2) = 22143375000 is the second smallest z such that z^8 = x^6 + y^7 for some x, y > 1, and the smallest corresponding y is a(2) = 30*z = 664301250000. It then follows that x = (22143375000^8 - 664301250000^7)^(1/6) = 29893556250000 = 1350*z.
Similarly, a(3) = 30*A300568(2) = 699840000000 is the smallest y for which x = (A300568(3)^8 - y^7)^(1/6) is an integer, here x = 1800*A300568(3) = 60*a(3).
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