A302174 -id:A302174 - cp's OEIS frontend

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

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

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

A300566 Numbers z such that there is a solution to x^4 + y^5 = z^6 with x, y, z >= 1.

Original entry on oeis.org

8748, 10368, 342732

Offset: 1

M. F. Hasler, Apr 16 2018

Also in the sequence: 810000 = 2^4*3^4*5^4, 1361367 = 3^4*7^5, 3240000 = 2^6*3^4*5^4, 9335088 = 2^4*3^5*7^4, 25312500 = 2^2*3^4*5^7, 31505922 = 2*3^8*7^4, 43740000 = 2^5*3^7*5^4, 512578125 = 3^8*5^7, 1215000000 = 2^6*3^5*5^7, 1701708750 = 2*3^4*5^4*7^5, 2196150000 = 2^4*3*5^5*11^4, 2431012500 = 2^2*3^4*5^5*7^4, 4269246912 = 2^6*3^4*7^7, 4447203750 = 2*3^5*5^4*11^4, 36015000000 = 2^6*3*5^7*7^4, 48717927500 = 2^2*5^4*11^7, 75969140625 = 3^4*5^8*7^4, 91116682272 = 2^5*3^4*7^4*11^4. - Jacques Tramu, Apr 17 2018
Consider a solution (x,y,z) of x^4 + y^5 = z^6. For any m, (x*m^15, y*m^12, z*m^10) is also a solution. Reciprocally, if (x/m^15, y/m^12, z/m^10) 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. - M. F. Hasler, Apr 17 2018
Observation: a(n) = A054744(n+38) = A257999(n+32), at least for 1 <= n <= 2 in both cases. - Omar E. Pol, Apr 17 2018
These relations hold only for n = 1 and 2. The next larger known term 342732 = 2^2*3*13^4 shows that in general the terms don't belong to A054744 nor A257999, although the earlier comment implies that each term gives rise to infinitely many non-primitive terms in A054744. - M. F. Hasler, Apr 19 2018
When S = a^4 + b^10/4 is a square, then z = b^5/2 + sqrt(S) is a solution, with x = a*z and y = b*z. All known solutions and further solutions 8957952, 10616832, 52200625, 216486432, ... are of this form (with rational a, b). - M. F. Hasler, Apr 19 2018

			a(1) = 8748 = 2^2*3^7 is in the sequence because 8748^6 = (2^3*3^8)^5 + (2^3*3^10)^4, using 2^3 + 1 = 3^2. Similarly, all z = 4*3^(10k-3) are in the sequence for k >= 1, with x = 8*3^(15k-5) and y = 8*3^(12k-4).
a(2) = 10368 = 2^7*3^4 is in the sequence because 10368^6 = (2^8*3^5)^5 + (2^10*3^6)^4, using 3 + 1 = 2^2. Similarly, any z = 2^7*3^(10k+4) is in the sequence for k >= 0, with x = 2^10*3^(15k+6) and y = 2^8*3^(12k+5).
z = 342732 = 2^2*3*13^4 is in the sequence because (2^2*3*13^4)^6 = (2^3*13^5)^5 + (2^3*5*13^6)^4, using 2^3*13 + 5^4 = 3^6.
z = 810000 = 2^4*3^4*5^4 is in the sequence because z^6 = x^4 + y^5 with x = 2^5*3^6*5^6 and y = 2^4*3^5*5^5 (using 1 + 3*5 = 2^4).
z = 1361367 = 3^4*7^5 is in the sequence because z^6 = x^4 + y^5 with x = 3^5*7^8 and y = 2*3^4*7^6.

Cf. A300564 (z^4 = x^2 + y^3) and A242183, A300565 (z^5 = x^3 + y^4), A302174.
Cf. A100294: numbers of the form a^5 + b^4.
See A303266 for the y-values.

is(z)=for(y=1,sqrtnint(-1+z=z^6,5),ispower(z-y^5,4)&&return(y))
/* Code below for illustration only, not guaranteed to give a complete list. Half-integral values give the additional term 31505922 for b = 63/2. Third-integral values give the additional solution z = 342732 for b = 26/3. */
S=[]; N=1e5; forstep(b=1,9,1/3, forstep(a=1,N,1/3, issquare(b^10+a^4<<2,&r)&& !frac(z=b^5/2+r/2)&& !print1(z",")&&S=setunion(S,[z])); print1([b])); S

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.

A302859 Primes of the form (k+1)!*k/2 + 1.

Original entry on oeis.org

2, 7, 37, 241, 1801, 15121, 141121, 1451521, 16329601, 199584001, 37362124801, 566658892801, 9153720576001, 23112569077678080001, 186134520519971831808000001

Offset: 1

Maheswara Rao Valluri, Apr 03 2018

nonn

The next term, for k = 251 (see A301373), is
2566282033898537172673689833660299199318441\
    47812028978290091772271674111238846647249346322032725585967946013083615\
    44220440938904033673583084158870025082998875790404475054647299641196409\
    72934112662249702715026203933143550590243427364871765801696382591273000\
    77256511620017707120387621962694782616283336623216978502662268159966484\
    36506095391239127788493879085200485514817503469381297494013097308996216\
    58710310236069486145497777789215839354880000000000000000000000000000000\
    0000000000000000000000000000001

Maheswara Rao Valluri, Primes of the form p = 1 + n! Sum n, for some n ∈ N*, arXiv:1803.11461 [math.GM], 2018.

Cf. A300559, A301373.

Reap[For[k = 1, k <= 1000, k++, If[PrimeQ[p = (k+1)! k/2 + 1], Print["k = ", k, " p = ", p]; Sow[p]]]][[2, 1]]

a(n) = A300559(A301373(n)) for all n >= 1; a(n) = A300559(n) for 1 <= n <= 10. - M. F. Hasler, Apr 15 2018

This sequence was originally submitted as A302174, then withdrawn, then reinstated with a new A-number by N. J. A. Sloane, Apr 14 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

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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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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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A300566 Numbers z such that there is a solution to x^4 + y^5 = z^6 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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A302859 Primes of the form (k+1)!*k/2 + 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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