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%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