A300566 -id:A300566 - cp's OEIS frontend

A303266 Least y for which x^4 + y^5 = z^6 for some x > 1 and z = A300566(n).

52488, 62208, 2970344

Offset: 1

M. F. Hasler, Apr 23 2018

See the main entry A300566 for all further information.

The values listed here are the y-values corresponding to the z-values listed in A300566. The x-values are then readily computed as (z^6 - y^5)^(1/4).

			A300566(1) = 8748 is the smallest z such that z^6 = x^4 + y^5 for some x,y > 1, and the smallest such y is a(1) = 52488. It then follows that x = (8748^6 - 52488^5)^(1/4) = 472392.
A300566(2) = 10368 is the second smallest z such that z^6 = x^4 + y^5 for some x, y > 1, and the smallest corresponding y is a(2) = 62208. It then follows that x = (10368^6 - 62208^5)^(1/4) = 746496.

A100291 Numbers of the form a^4 + b^3 with a, b > 0.

Original entry on oeis.org

2, 9, 17, 24, 28, 43, 65, 80, 82, 89, 108, 126, 141, 145, 206, 217, 232, 257, 264, 283, 297, 320, 344, 359, 381, 424, 472, 513, 528, 593, 599, 626, 633, 652, 689, 730, 745, 750, 768, 810, 841, 968, 985, 1001, 1016, 1081, 1137, 1256, 1297, 1304, 1323, 1332

Offset: 1

T. D. Noe, Nov 18 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Gian Cordana Sanjaya and Xiaoheng Wang, On the squarefree values of a^4+b^3, arXiv:2107.10380 [math.NT], 2021.

Cf. A100271 (primes of the form a^4 + b^3).
Cf. A055394 (a^2 + b^3: contains this as subsequence), 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), A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).
Roots of 5th powers are listed in A300565 (z^5 = x^3 + y^4); see also A300564 (z^4 = x^2 + y^3) and A242183, A300566 (z^6 = x^4 + y^5), A300567 (z^7 = x^6 + y^5), A302174.

lst={}; Do[p=a^4+b^3; If[p<2000, AppendTo[lst, p]], {a, 64}, {b, 256}]; Union[lst]
With[{nn=20},Select[Union[#[[1]]^4+#[[2]]^3&/@Tuples[Range[20],2]],#<= nn^3+1&]] (* Harvey P. Dale, May 27 2020 *)

is(n)=for(a=1, sqrtnint(n-1, 4), ispower(n-a^4, 3) && return(a)) \\ Returns a > 0 if n is in the sequence, or 0 otherwise. - M. F. Hasler, Apr 25 2018

list(lim)=my(v=List());for(b=1,sqrtnint(lim\=1,3), my(b3=b^3); for(a=1,sqrtnint(lim-b3,4), listput(v,a^4+b3))); Set(v) \\ Charles R Greathouse IV, Jul 26 2021

Edited by M. F. Hasler, Apr 25 2018

A100294 Numbers of the form a^5 + b^4 with a, b > 0.

Original entry on oeis.org

2, 17, 33, 48, 82, 113, 244, 257, 259, 288, 324, 499, 626, 657, 868, 1025, 1040, 1105, 1280, 1297, 1328, 1539, 1649, 2320, 2402, 2433, 2644, 3126, 3141, 3206, 3381, 3425, 3750, 4097, 4128, 4339, 4421, 5120, 5526, 6562, 6593, 6804, 7221, 7585, 7777, 7792

Offset: 1

T. D. Noe, Nov 18 2004

nonn

In view of computing A300566, it would be interesting to have an efficient way to check whether a given (large) n is in this sequence. - M. F. Hasler, Apr 25 2018

Cf. A100274 (primes of the form a^5 + b^4).
Subsequence of A100292 (a^5 + b^2); see also A055394 (a^2 + b^3), A111925 (a^2 + b^4), A100291 (a^4 + b^3), A100293 (a^5 + b^3), A303372 (a^2 + b^6), A303373 (a^3 + b^6), A303374 (a^4 + b^6), A303375 (a^5 + b^6).
Roots of 6th powers are listed in A300566 (z such that z^6 = x^5 + y^4 for some x, y >= 1); see also A300564 (z^4 = x^3 + y^2) and A242183, A300565 (z^5 = x^4 + y^3), A300567 (z^7 = x^6 + y^5), A302174.

lst={}; Do[p=a^5+b^4; If[p<15000, AppendTo[lst, p]], {a, 16}, {b, 32}]; Union[lst]

A100294_vec(L=10^6, k=4, m=5, 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. - M. F. Hasler, Apr 25 2018

is(n, k=4, m=5)=for(a=1, sqrtnint(n-1, m), ispower(n-a^m,k) && return(a)) \\ Returns a > 0 if n is in the sequence, or 0 otherwise. - M. F. Hasler, Apr 25 2018

A300564 Numbers z such that there is a solution to x^2 + y^3 = z^4 with x, y, z >= 1.

Original entry on oeis.org

6, 9, 15, 35, 36, 42, 48, 57, 63, 71, 72, 75, 78, 90, 98, 100, 120, 135, 141, 147, 162, 195, 196, 204, 208, 215, 225, 243, 252, 260, 279, 280, 288, 289, 295, 300, 306, 336, 363, 364, 384, 405, 441, 450, 456, 462, 504, 510, 525, 537, 550, 568, 576, 600, 624, 630, 713, 720, 722, 735, 750, 784, 800, 819, 828, 841, 845, 847, 867, 875

Offset: 1

M. F. Hasler, Apr 16 2018

nonn

Cf. A242183, A242192, A300565 (z^5 = x^3 + x^4), A300566 (z^6 = x^4 + y^5).

is(z)=for(y=1,sqrtnint(z^4,3),issquare(z^4-y^3,&x)&&x&&return(1))

Equals sequence A242183 with duplicates removed.

A300565 Numbers z such that there is a solution to x^3 + y^4 = z^5 with x, y, z >= 1.

Original entry on oeis.org

32, 250, 1944, 2744, 3888, 19208, 27648, 55296, 59049, 59582, 81000, 82944, 131072, 135000, 185193, 200000

Offset: 1

M. F. Hasler, Apr 16 2018

Consider a solution (x, y, z), x^3 + y^4 = z^5. For any m, (x*m^20, y*m^15, z*m^12) will also be a solution. If (x/m^20, y/m^15, z/m^12) is a triple of integers, it is also a solution. A solution is called primitive if there is no such m > 1.

If S = a^3 + b^8/4 is a square, for some a,b > 0, then z = b^4/2 + sqrt(S) is in the sequence, with x = a*z and y = b*z. All known terms are of this form, with b in {2, 6, 7, 9, 12}, only for a(2) and a(10) one must consider half-integral b = 5/2 resp. 31/2. Also of this form is z = 81000, 82944, 131072, 135000, 185193, 200000, 243000, 395307, 474552, 574992, 800000, 820125, 862488, 864000, 972000, ... (with integer b), and 444528 (with b = 33/2).

			a(1) = 32 = 2^5 is in the sequence because (2^5)^5 = (2^6)^4 + (2^8)^3, using 1 + 1 = 2.
a(2) = 250 = 2*5^3 is in the sequence because 250^5 = 2^5*5^15 = (5^4)^4 + (3*5^5)^3, using 5 + 3^3 = 2^5. This solution is special because x and y are not multiples of z.
a(3) = 1944 = 2^3*3^5 is in the sequence because 1944^5 = (2^4*3^6)^4 + (2^5*3^8)^3, using 2 + 1 = 3.
a(7) = 27648 = 2^10*3^3 is in the sequence because (2^10*3^3)^5 = (2^12*3^4)^4 + (2^16*3^5)^3, using 3 + 1 = 2^2.
a(10) = 59582 = 2*31^3 is in the sequence because (2*31^3)^5 = (31^4)^4 + (31^5)^3, using 31 + 1 = 2^5. This is the second case where x and y are not multiples of z.

Cf. A300564 (z^4 = x^2 + y^3), A111925 (z^5 = x^2 + y^4), A302174, A300566 (z^6 = x^4 + y^5).

is(z)=for(y=1,sqrtnint(-1+z=z^5,4),ispower(z-y^4,3)&&return(y))

a(11)-a(16) from Giovanni Resta, Jan 23 2020

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

M. F. Hasler, Apr 16 2018

Also in the sequence: 72900000 = 2^5*3^6*5^5, 51018336 = 2^5*3^13, 6083264512 = 2^14*13^5, 916132832 = 2^5*31^5, 6530347008 = 2^12*3^13, 16307453952 = 2^26*3^5, 233861123808 = 2^5*3^9*13^5, 850305600000 = 2^9*3^12*5^5.
Consider a solution (x,y,z) of x^5 + y^6 = z^7. For any m, (x*m^42, y*m^35, z*m^30) is also a solution. Reciprocally, if (x/m^42, y/m^35, z/m^30) 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^5 + b^12/4 is a square, then z = b^6/2 + sqrt(S) is a solution, with x = a*z and y = b*z. All known solutions are of this form. Sequence A303267 lists the y-values, thus equal to b*z where z = a(n), with corresponding x = a*z = (a(n)^7 - A303267(n)^6)^(1/5).

			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.

Hayden Chesnut, A300567 Python Code

Cf. A300564 (z^4 = x^2 + y^3) and A242183, A300565 (z^5 = x^3 + y^4), A300566 (z^6 = x^4 + y^5), A302174.
See A303267 for the y-values.
Cf. A303375 (numbers of the form a^5 + b^6).

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

# See Hayden link. This code is built to identify valid z values based on specific conjectures outlined in the file.

A302174 Smallest solution x of x^n + y^(n+1) = z^(n+2), x, y, z >= 1.

Original entry on oeis.org

1, 2, 27, 256, 472392, 262144, 13759414272, 4294967296, 4057816381784064

Offset: 0

Jacques Tramu, Apr 07 2018

From M. F. Hasler, Apr 13 2018: (Start)
Proofs for the upper limits given in the formula section:
For odd n = 2k-1, x = 2^(2*k^2) yields a solution with x^(2k-1) = y^(2k) = 1/2*z^(2k+1), y = 2^(k(2k-1)) and z = 2^(2k(k-1)+1), because 2*k^2*(2k-1) + 1 = (2k+1)*(2k(k-1)+1).
For even n = 2k, x = 2^(k*(2k+1))*3^(2k+2) yields a solution, with y = 2^(2*k^2)*3^(2k+1) and z = 2^(2*k^2-k+1)*3^(2k), because for the exponents of 3, (2k+1)^2 = (2k+2)2k + 1 and the factor 1+3 = 2^2 adds 2 to the (identical) exponent of 2 in x^(2k) and y^(2k+1), to factor as 2*k^2(2k+1) + 2 = (2k+2)(k(2k-1)+1). (End)

			1^0 + 3^1 = 2^2, therefore a(0) = 1.
2^1 + 5^2 = 3^3, so a(1) = 2. (No solution can have x = 1 because z^3 - 1 = (z - 1)(z^2 + z + 1) cannot be a square: if a = z - 1, then z^2 + z + 1 = a^2 + 3a + 3 is congruent to 3 modulo any factor of a, and a = 3b yields z^3 - 1 = 9*b*(3*b^2 + 3b + 1), the last factor being congruent to 1 modulo any factor of b, and cannot be a square.)
27^2 + 18^3 = 9^4, so a(2) = 27.
256^3 + 64^4 = 32^5, so a(3) = 256.
472392^4 + 52488^5 = 8748^6, so a(4) = 472392.

Cf. A001105 (2*k^2), A060757 (4^k^2 = 2^(2k^2)), A000244 (3^k).
Conjectured to be a subsequence of A003586 (2^i*3^j).
Cf. A300564, A300565, A300566, A300567, A300568 (z^4 = x^2 + y^3, ..., z^8 = x^6 + y^7).

For odd n, a(n) <= 2^((n+1)^2/2); for even n, a(n) <= 2^(n*(n+1)/2)*3^(n+2).

We may conjecture that, for n > 4, a(n) is given by these upper limits.

Extended to a(0) = 1 by M. F. Hasler, Apr 13 2018

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

Original entry on oeis.org

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

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

M. F. Hasler, May 04 2018

nonn

The values listed here are the y-values corresponding to the z-values listed in A300568. The x-values are then readily computed as (z^8 - y^7)^(1/6).

See the main entry A300567 for all further information.

			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).

Cf. A300564, A300565, A300566, A300567, A300568, A303265, A303266, A303267.

cp's OEIS Frontend

A303266 Least y for which x^4 + y^5 = z^6 for some x > 1 and z = A300566(n).

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A100291 Numbers of the form a^4 + b^3 with a, b > 0.

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A100294 Numbers of the form a^5 + b^4 with a, b > 0.

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A300564 Numbers z such that there is a solution to x^2 + y^3 = z^4 with x, y, z >= 1.

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A300565 Numbers z such that there is a solution to x^3 + y^4 = z^5 with x, y, z >= 1.

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A300567 Numbers z such that z^7 = x^5 + y^6 for some integers x, y >= 1.

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A302174 Smallest solution x of x^n + y^(n+1) = z^(n+2), x, y, z >= 1.

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

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A303268 Least y for which x^6 + y^7 = A300568(n)^8 for some x > 1.

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