This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A359314 #45 Dec 10 2023 09:12:25
%S A359314 3,19,22,10,15,23,15,52,65,36,37,67,23,54,73,33,47,74,3,55,80,32,43,
%T A359314 81,11,65,78,37,50,81
%N A359314 Three-column table T(n,k) read by rows where the elements in the pair of two adjacent rows, starting with the odd-indexed row T(2j-1,k) and followed by the even-indexed one T(2j,k), are such that they are not multiples of the elements presented in the previous rows and that Sum_{k=1..3} T(2j-1,k)^2 = Sum_{k=1..3} T(2j,k)^2 and Sum_{k=1..3} T(2j-1,k)^6 = Sum_{k=1..3} T(2j,k)^6 for j > 0 and k = 1, 2, 3.
%C A359314 It was found empirically (via computer calculations) that for integers a, b, c, d, e and f satisfying a^6 + b^6 + c^6 = d^6 + e^6 + f^6, it is also most likely to be true that a^2 + b^2 + c^2 = d^2 + e^2 + f^2.
%C A359314 Such cases are presented in this sequence where
%C A359314 a = T(2j-1,1), b = T(2j-1,2) c = T(2j-1,3) and
%C A359314 d = T(2j,1), e = T(2j,2), f = T(2j,3).
%C A359314 There currently exists no formula to calculate terms of this sequence -- they have to be found via trial and test (computer) calculations.
%C A359314 Each row consists of 3 columns.
%C A359314 The table starts with the rows which have the smallest sums of squares of elements (such sums also correspond to the smallest sums of the same 6th powers of the same elements) -- see the EXAMPLE section. The terms in each row are presented in ascending order.
%D A359314 R. K. Guy, Unsolved problems in Number theory, chapter D, section D1, page 213.
%H A359314 Simcha Brudno, <a href="https://doi.org/10.1090/S0025-5718-1976-0406923-6">Triples of 6th powers with equal sums</a>, Mathematics of Computation, vol. 30, nb. 135, July 1976.
%H A359314 Bernard Montaron, <a href="https://www.quora.com/How-can-we-prove-that-if-six-natural-integers-are-such-that-p-6-q-6-r-6-u-6-v-6-w-6-then-p-2-q-2-r-2-u-2-v-2-w-2">How can we prove that if six natural integers are such that p^6+q^6+r^6=u^6+v^6+w^6 then p^2+q^2+r^2=u^2+v^2+w^2?</a>, Quora.
%e A359314 Table begins:
%e A359314 k=1 k=2 k=3 SquaresSum 6thPowersSum
%e A359314 n=1: 3, 19, 22; 854 160426514
%e A359314 n=2: 10, 15, 23; 854 160426514
%e A359314 n=3: 15, 52, 65; 7154 95200890914
%e A359314 n=4: 36, 37, 67; 7154 95200890914
%e A359314 n=5: 23, 54, 73; 8774 176277173474
%e A359314 n=6: 33, 47, 74; 8774 176277173474
%e A359314 n=7: 3, 55, 80; 9434 289824641354
%e A359314 n=8: 32, 43, 81; 9434 289824641354
%e A359314 n=9: 11, 65, 78; 10430 300620262890
%e A359314 n=10: 37, 50, 81; 10430 300620262890
%e A359314 ...
%e A359314 The elements of the row n=1: 3, 19, 22 and the elements of the row n=2: 10, 15, 23 are such that 3^2 + 19^2 + 22^2 = 10^2 + 15^2 + 23^2 and 3^6 + 19^6 + 22^6 = 10^6 + 15^6 + 23^6.
%K A359314 nonn,tabf,more
%O A359314 1,1
%A A359314 _Alexander R. Povolotsky_, Dec 25 2022