A100293 -id:A100293 - cp's OEIS frontend

A274027 Numbers k such that k^4 is the average of a positive cube and a positive fifth power.

1, 162, 324, 3888, 11664, 18750, 31250, 32768, 38416, 40000, 160000, 167042, 168750, 253125, 373248, 607500, 911250, 1037232

Offset: 1

Altug Alkan, Jun 07 2016

Numbers n such that 2*n^4 is of the form x^3 + y^5 where x and y are positive integers.

Sequence is infinite because if m is a term, that is m^4 = (w^3 + z^5)/2 with w and z positive integers, then m*t^15 is also a term for every integer t>1. In fact: (m*t^15)^4 = ((w*t^20)^3 + (z*t^12)^5)/2.

			162 = 3*54 is a term because (3*54)^4 = ((18*54)^3 + 54^5)/2.
38416 = 14^4 is a term because (14^4)^4 = ((3*14^5)^3 + (14^3)^5)/2.

Cf. A000583, A100293.

isA100293(n) = for(y=1, sqrtnint(n-1, 5), if(ispower(n-y^5, 3), return(1))); 0;
lista(nn) = for(n=1, nn, if(isA100293(2*n^4), print1(n, ", ")));

a(11)-a(18) from Giovanni Resta, Jun 07 2016

A274033 Numbers k such that k = a^2 + b^4 and n^2 = c^3 + d^5 for some positive integers a, b, c, d.

Original entry on oeis.org

81250, 1062882, 11529602, 12500000, 170061120, 200000000, 2662400000, 5897400777, 7309688832, 12814453125, 34297420960, 37019531250

Offset: 1

Altug Alkan, Jun 07 2016

In other words, values of a^2 + b^4 such that (a^2 + b^4)^2 is of the form c^3 + d^5 where a, b, c, d > 0.
81250 is the least number with this property.
Sequence is infinite: If n = a^2 + b^4 and n^2 = c^3 + d^5, then n*k^60 = (a*k^30)^2 + (b*k^15)^4 and (n*k^60)^2 = (c*k^40)^3 + (d*k^24)^5. So if n is in this sequence, then n*k^60 is in this sequence for all nonzero values of k.

			81250 is a term because 81250 = 175^2 + 15^4 and 81250^2 = 1875^3 + 25^5.

Cf. A100293, A111925.

a(2)-a(6) from Giovanni Resta, Jun 07 2016
a(7) from Chai Wah Wu, Jun 14 2016
a(8)-a(12) from Chai Wah Wu, Jul 07 2016

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

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A274027 Numbers k such that k^4 is the average of a positive cube and a positive fifth power.

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A274033 Numbers k such that k = a^2 + b^4 and n^2 = c^3 + d^5 for some positive integers a, b, c, d.

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

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