A111925 -id:A111925 - cp's OEIS frontend

A274367 Taxi-cab numbers (A001235) that are of the form x^2 + y^4 in more than one way (x, y > 0).

27445392, 1644443281, 2367885312, 5687433577, 112416325632, 208265121792, 900069054976, 1976398601697, 6735639678976, 9698858237952, 9911785815477, 14585606569872, 15283760730112, 18156501172017, 23295727931392, 29871321586561, 33510832422912, 67250060669952

Offset: 1

Altug Alkan, Jun 19 2016

nonn

A272701(3) = 27445392 is the least number with the property that sequence focuses on.

If n = a^3 + b^3 = c^3 + d^3 = x^2 + y^4 = z^2 + t^4, then n*k^12 = (a*k^4)^3 + (b*k^4)^3 = (c*k^4)^3 + (d*k^4)^3 = (x*k^6)^2 + (y*k^3)^4 = (z*k^6)^2 + (t*k^3)^4. So if n is this sequence, then n*k^12 is also in this sequence for all k > 1.

			27445392 is a term because 27445392 = 141^3 + 291^3 = 198^3 + 270^3 = 756^2 + 72^4 = 5076^2 + 36^4.
112416325632 is a term because 112416325632 = 27445392*2^12.

Cf. A001235, A111925, A272701.

a(2)-a(18) from Giovanni Resta, Jun 19 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

cp's OEIS Frontend

A274367 Taxi-cab numbers (A001235) that are of the form x^2 + y^4 in more than one way (x, y > 0).

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