A274219
Numbers n such that n^k is the average of a positive square and a positive cube for all k > 0.
Original entry on oeis.org
1, 64, 729, 1000, 1728, 4096, 13456, 15625, 27000, 46656, 59319, 64000, 110592, 117649, 148877, 262144, 531441, 729000, 777924, 861184, 1000000, 1259712, 1537600, 1728000
Offset: 1
13456 = 116^2 is a term because 13456 = (163^2 + 7^3) / 2, 13456^2 = (1088^2 + 712^3) / 2, 13456^3 = ((116^3)^2 + (116^2)^3) / 2, 13456^4 = (254426048^2 + 94192^3) / 2, 13456^5 = (1698254848^2 + 9580672^3) / 2, 13456^6 = ((116^6)^2 + (116^4)^3) / 2.
A274247
Numbers n such that n^k is the sum of a positive square and a positive cube for all k not divisible by 6.
Original entry on oeis.org
12348, 16464, 433664, 444528, 617400, 790272, 1053696, 2534400, 2737152, 6585600, 6667920, 7024032
Offset: 1
12348 is a term since 12348 = 98^2 + 14^3, 12348^2 = 9604^2 + 392^3, 12348^3 = 1361367^2 + 3087^3, 12348^4 = 76236552^2 + 259308^3, 12348^5 = 11206773144^2 + 5445468^3.
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
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, ...
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With[{nn=40}, Take[Union[First[#]^7 + Last[#]^8&/@Tuples[Range[nn], 2]], nn]]
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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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