login
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.
0
3, 19, 22, 10, 15, 23, 15, 52, 65, 36, 37, 67, 23, 54, 73, 33, 47, 74, 3, 55, 80, 32, 43, 81, 11, 65, 78, 37, 50, 81
OFFSET
1,1
COMMENTS
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.
Such cases are presented in this sequence where
a = T(2j-1,1), b = T(2j-1,2) c = T(2j-1,3) and
d = T(2j,1), e = T(2j,2), f = T(2j,3).
There currently exists no formula to calculate terms of this sequence -- they have to be found via trial and test (computer) calculations.
Each row consists of 3 columns.
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.
REFERENCES
R. K. Guy, Unsolved problems in Number theory, chapter D, section D1, page 213.
EXAMPLE
Table begins:
k=1 k=2 k=3 SquaresSum 6thPowersSum
n=1: 3, 19, 22; 854 160426514
n=2: 10, 15, 23; 854 160426514
n=3: 15, 52, 65; 7154 95200890914
n=4: 36, 37, 67; 7154 95200890914
n=5: 23, 54, 73; 8774 176277173474
n=6: 33, 47, 74; 8774 176277173474
n=7: 3, 55, 80; 9434 289824641354
n=8: 32, 43, 81; 9434 289824641354
n=9: 11, 65, 78; 10430 300620262890
n=10: 37, 50, 81; 10430 300620262890
...
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.
CROSSREFS
Sequence in context: A185446 A172032 A043073 * A022128 A041381 A042231
KEYWORD
nonn,tabf,more
AUTHOR
STATUS
approved