cp's OEIS Frontend

A308503 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.

%I A308503 #10 Jun 03 2019 09:14:28
%S A308503 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 A308503 7,8,77,42,245,28,105,22,8,9,24,91,150,65,60,39,54,9,10,45,40,847,66,
%U A308503 385,44,375,50,10,11,70,63,72,119,210,85,108,147,42,11,12
%N A308503 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 A308503 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 A308503 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 A308503 The function g is defined by:
%C A308503 - g(1) = () (the empty sequence),
%C A308503 - g(n) = the n-th row of A067255 for any n > 1.
%F A308503 For any m, n, k > 0:
%F A308503 - T(m, T(n, k)) = T(T(m, n), k) (T is associative),
%F A308503 - T(n, 1) = T(1, n) = n (1 is a neutral element),
%F A308503 - T(2, k) = 2*A003961(k),
%F A308503 - h(T(n, k)) = h(n) + h(k) for h = A001221, A001222, A061395,
%F A308503 - the function i -> T(n, i)/n is completely multiplicative and equals the A061395(n)-th iterate of A003961.
%e A308503 Array T(n, k) begins:
%e A308503   n\k|   1   2    3    4    5     6    7     8     9    10
%e A308503   ---+----------------------------------------------------
%e A308503     1|   1   2    3    4    5     6    7     8     9    10
%e A308503     2|   2   6   10   18   14    30   22    54    50    42
%e A308503     3|   3  15   21   75   33   105   39   375   147   165
%e A308503     4|   4  12   20   36   28    60   44   108   100    84
%e A308503     5|   5  35   55  245   65   385   85  1715   605   455
%e A308503     6|   6  30   42  150   66   210   78   750   294   330
%e A308503     7|   7  77   91  847  119  1001  133  9317  1183  1309
%e A308503     8|   8  24   40   72   56   120   88   216   200   168
%e A308503     9|   9  45   63  225   99   315  117  1125   441   495
%e A308503    10|  10  70  110  490  130   770  170  3430  1210   910
%o A308503 (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 A308503 Cf. A001221, A001222, A003961, A061395, A067255.
%K A308503 nonn,tabl
%O A308503 1,2
%A A308503 _Rémy Sigrist_, Jun 02 2019