A308503 - OEIS

%I #10 Jun 03 2019 09:14:28

%S 1,2,2,3,6,3,4,15,10,4,5,12,21,18,5,6,35,20,75,14,6,7,30,55,36,33,30,

%T 7,8,77,42,245,28,105,22,8,9,24,91,150,65,60,39,54,9,10,45,40,847,66,

%U 385,44,375,50,10,11,70,63,72,119,210,85,108,147,42,11,12

%N Square array T(n, k), n, k > 0, read by antidiagonals upwards: T(n, k) = f(g(n) | g(k)), where f is defined over the set of finite sequences of nonnegative integers with no trailing zero as f(e) = Sum_{k = 1..#e} prime(k)^e_k, g is the inverse of f, and | denotes concatenation.

%C The function f establishes a natural bijection from the set of finite sequences of nonnegative integers with no trailing zero to the set of natural numbers based on prime factorization.

%C If we consider the set of finite sequences of signed integers with no trailing zero, then we obtain a bijection to the set of positive rational numbers.

%C The function g is defined by:

%C - g(1) = () (the empty sequence),

%C - g(n) = the n-th row of A067255 for any n > 1.

%F For any m, n, k > 0:

%F - T(m, T(n, k)) = T(T(m, n), k) (T is associative),

%F - T(n, 1) = T(1, n) = n (1 is a neutral element),

%F - T(2, k) = 2*A003961(k),

%F - h(T(n, k)) = h(n) + h(k) for h = A001221, A001222, A061395,

%F - the function i -> T(n, i)/n is completely multiplicative and equals the A061395(n)-th iterate of A003961.

%e Array T(n, k) begins:

%e   n\k|   1   2    3    4    5     6    7     8     9    10

%e   ---+----------------------------------------------------

%e     1|   1   2    3    4    5     6    7     8     9    10

%e     2|   2   6   10   18   14    30   22    54    50    42

%e     3|   3  15   21   75   33   105   39   375   147   165

%e     4|   4  12   20   36   28    60   44   108   100    84

%e     5|   5  35   55  245   65   385   85  1715   605   455

%e     6|   6  30   42  150   66   210   78   750   294   330

%e     7|   7  77   91  847  119  1001  133  9317  1183  1309

%e     8|   8  24   40   72   56   120   88   216   200   168

%e     9|   9  45   63  225   99   315  117  1125   441   495

%e    10|  10  70  110  490  130   770  170  3430  1210   910

%o (PARI) T(n,k) = { my (e=concat(apply(m -> my (f=factor(m), w=#f~, v=vector(if (w, primepi(f[w,1]), 0))); for (j=1, w, v[primepi(f[j,1])]=f[j,2]); v, [n,k]))); prod (i=1, #e, if (e[i], prime(i)^e[i], 1)) }

%Y Cf. A001221, A001222, A003961, A061395, A067255.

%K nonn,tabl

%O 1,2

%A _Rémy Sigrist_, Jun 02 2019