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 A324592 #14 Dec 10 2023 17:52:23
%S A324592 1,1,1,1,2,1,1,3,3,1,1,4,4,4,1,1,5,9,9,5,1,1,6,11,16,11,6,1,1,7,12,25,
%T A324592 25,12,7,1,1,8,17,36,8,36,17,8,1,1,9,27,49,55,55,49,27,9,1,1,10,16,64,
%U A324592 31,72,31,64,16,10,1,1,11,33,81,125,119,119,125
%N A324592 Square array T(n, k) read by diagonals, n > 0, k > 0; for any number m > 0 with prime factorization Product_{i > 0} prime(i)^e(i), let f(m) = Sum_{i > 0} e(i) * sqrt(A005117(i)); f establishes a bijection between the positive numbers and the finite sums of square roots of squarefree numbers; let g be the inverse of f; T(n, k) = g(f(n) * f(k)).
%C A324592 The set of square roots of squarefree numbers, { sqrt(A005117(i)), i > 0 }, is Q-linearly independent. The set of finite sums of square roots of squarefree numbers is closed under multiplication, hence the sequence is well defined.
%C A324592 The function f can be naturally extended to the set of positive rational numbers: if r = u/v (not necessarily in reduced form), then f(r) = f(u) - f(v).
%C A324592 This sequence has similarities with A297845.
%H A324592 Eric Jaffe, <a href="http://www.math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALAPP/Jaffe.pdf">Linearly Independent Integer Roots over the Scalar Field Q</a>
%H A324592 Rémy Sigrist, <a href="/A324592/a324592.gp.txt">PARI program for A324592</a>
%F A324592 For any m > 0, n > 0 and k > 0:
%F A324592 - T(n, k) = T(k, n) (T is commutative),
%F A324592 - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F A324592 - T(m, n*k) = T(m, n) * T(m, k) and T(n*k, m) = T(n, m) * T(k, m) (T is completely multiplicative in both parameters),
%F A324592 - T(n, 1) = 1 (1 is an absorbing element for T),
%F A324592 - T(n, 2) = n (2 is an identity element for T),
%F A324592 - T(n, 2^i) = n^i for any i >= 0,
%F A324592 - A001221(T(n, k)) <= A001221(n) * A001221(k),
%F A324592 - T(prime(n), prime(n)) = 2^A005117(n) (where prime(n) denotes the n-th prime number).
%e A324592 Array T(n, k) begins:
%e A324592 n\k| 1 2 3 4 5 6 7 8 9 10
%e A324592 ---+-------------------------------------------------
%e A324592 1| 1 1 1 1 1 1 1 1 1 1
%e A324592 2| 1 2 3 4 5 6 7 8 9 10
%e A324592 3| 1 3 4 9 11 12 17 27 16 33
%e A324592 4| 1 4 9 16 25 36 49 64 81 100
%e A324592 5| 1 5 11 25 8 55 31 125 121 40
%e A324592 6| 1 6 12 36 55 72 119 216 144 330
%e A324592 7| 1 7 17 49 31 119 32 343 289 217
%e A324592 8| 1 8 27 64 125 216 343 512 729 1000
%e A324592 9| 1 9 16 81 121 144 289 729 256 1089
%e A324592 10| 1 10 33 100 40 330 217 1000 1089 400
%e A324592 For n = 3 and k = 5:
%e A324592 - f(3) = f(prime(2)) = sqrt(A005117(2)) = sqrt(2),
%e A324592 - f(5) = f(prime(3)) = sqrt(A005117(3)) = sqrt(3),
%e A324592 - f(3) * f(5) = sqrt(6) = sqrt(A005117(5)),
%e A324592 - hence T(3, 5) = prime(5) = 11.
%o A324592 (PARI) See Links section.
%Y A324592 Cf. A001221, A005117, A297845.
%K A324592 nonn,tabl
%O A324592 1,5
%A A324592 _Rémy Sigrist_, Sep 03 2019