A303376 -id:A303376 - cp's OEIS frontend

A300568 Numbers z such that z^8 = x^6 + y^7 for some integers x, y >= 1.

47775744, 22143375000, 23328000000

Offset: 1

M. F. Hasler, May 04 2018

Also in the sequence: 4001504141376, 4275897935643, 11284439629824, 100192997081088, ... (with b = 63, 51, 72, 96, ... cf. below).
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.
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.
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).

			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.
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.

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.
See A303268 for the y-values.
Cf. A303376 (numbers of the form a^6 + b^7).

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.
/* Code below for illustration only, not guaranteed to give a complete list. */
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

A303377 Numbers of the form a^7 + b^8, with integers a, b > 0.

Original entry on oeis.org

2, 129, 257, 384, 2188, 2443, 6562, 6689, 8748, 16385, 16640, 22945, 65537, 65664, 67723, 78126, 78381, 81920, 84686, 143661, 279937, 280192, 286497, 345472, 390626, 390753, 392812, 407009, 468750, 670561, 823544, 823799, 830104, 889079, 1214168, 1679617, 1679744, 1681803

Offset: 1

M. F. Hasler, May 04 2018

Although it is easy to produce many terms of this sequence, it is nontrivial to check efficiently whether a very large number is of this form.

			The sequence starts with 1^7 + 1^8, 2^7 + 1^8, 1^7 + 2^8, 2^7 + 2^8, 3^7 + 1^8, 3^7 + 2^8, 1^7 + 3^8, 2^7 + 3^8, 3^7 + 3^8, 4^7 + 1^8, 4^7 + 2^8, 4^7 + 3^8, 1, ...

Cf. A000404 (a^2 + b^2), A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100292 (a^5 + b^2), A100293 (a^5 + b^3), A100294 (a^5 + b^4).

Cf. A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6), A303376 (a^6 + b^7).

With[{nn=40}, Take[Union[First[#]^7 + Last[#]^8&/@Tuples[Range[nn], 2]], nn]]

is(n,k=7,m=8)=for(b=1,sqrtnint(n-1,m),ispower(n-b^m,n)&&return(b)) \\ Returns b > 0 if n is in the sequence, else 0.
A303377_vec(L=10^7,k=7,m=8,S=List())={for(a=1,sqrtnint(L-1,m),for(b=1,sqrtnint(L-a^m,k), listput(S,a^m+b^k)));Set(S)} \\ all terms up to limit L

cp's OEIS Frontend

A300568 Numbers z such that z^8 = x^6 + y^7 for some integers x, y >= 1.

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