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%I #14 Jul 06 2019 02:20:13
%S 1,2,1,1,2,1,4,2,2,1,1,4,3,2,1,2,2,4,4,2,1,1,4,5,4,5,2,1,8,2,6,16,4,6,
%T 2,1,1,8,7,8,11,4,7,2,1,2,4,8,256,10,90,4,8,2,1,1,4,9,8,17,12,17,4,9,
%U 2,1,4,2,10,16,8,47250,14,512,4,10,2,1,1,8
%N Square array T(n, k) read by antidiagonals upwards, n > 0 and k > 0: for any number n > 0, let f(n) be the polynomial of a single indeterminate x where the coefficient of x^e is the prime(1+e)-adic valuation of n (where prime(k) denotes the k-th prime); f establishes a bijection between the positive numbers and the polynomials of a single indeterminate x with nonnegative integer coefficients; let g be the inverse of f; T(n, k) = g(f(n) o f(k)) (where o denotes function composition).
%C This sequence has connections with A297845.
%C 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); as such, f is a homomorphism from the multiplicative group of positive rational numbers to the additive group of polynomials of a single indeterminate x with integer coefficients.
%F For any m, n, k > 0 and any i >= 0:
%F - T(1, k) = 1,
%F - T(2^i, k) = 2^i,
%F - T(3, k) = k,
%F - T(3^i, k) = k^i,
%F - T(5, k) = A297473(k),
%F - T(6, k) = 2*k,
%F - T(n, 1) = A006519(n),
%F - T(n, 2) = A061142(n),
%F - T(n, 3) = n,
%F - T(n, 5) = A319525(n),
%F - T(m*n, k) = T(m, k) * T(n, k).
%e Array T(n, k) begins:
%e n\k| 1 2 3 4 5 6 7 8 9 10
%e ---+-----------------------------------------------------------
%e 1| 1 1 1 1 1 1 1 1 1 1
%e 2| 2 2 2 2 2 2 2 2 2 2
%e 3| 1 2 3 4 5 6 7 8 9 10
%e 4| 4 4 4 4 4 4 4 4 4 4
%e 5| 1 2 5 16 11 90 17 512 625 550
%e 6| 2 4 6 8 10 12 14 16 18 20
%e 7| 1 2 7 256 17 47250 29 134217728 5764801 5656750
%e 8| 8 8 8 8 8 8 8 8 8 8
%e 9| 1 4 9 16 25 36 49 64 81 100
%e 10| 2 4 10 32 22 180 34 1024 1250 1100
%e The corresponding polynomials are:
%e f(n)\f(k)| 0 1 x 2 x^2 x+1 x^3 3 2*x x^2+1
%e ---------+---------------------------------------------------------------------
%e 0| 0 0 0 0 0 0 0 0 0 0
%e 1| 1 1 1 1 1 1 1 1 1 1
%e x| 0 1 x 2 x^2 x+1 x^3 3 2*x x^2+1
%e 2| 2 2 2 2 2 2 2 2 2 2
%e x^2| 0 1 x^2 4 x^4 x^2+2*x+1 x^6 9 4*x^2 x^4+2*x^2+1
%e x+1| 1 2 x+1 3 x^2+1 x+2 x^3+1 4 2*x+1 x^2+2
%e x^3| 0 1 x^3 8 x^6 x^3+3*x^2+3*x+1 x^9 27 8*x^3 x^6+3*x^4+3*x^2+1
%e 3| 3 3 3 3 3 3 3 3 3 3
%e 2*x| 0 2 2*x 4 2*x^2 2*x+2 2*x^3 6 4*x 2*x^2+2
%e x^2+1| 1 2 x^2+1 5 x^4+1 x^2+2*x+2 x^6+1 10 4*x^2+1 x^4+2*x^2+2
%o (PARI) g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1))
%o f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i,2] * v^(primepi(f[i,1]) - 1))
%o T(n,k) = g(f(n, f(k)))
%Y See A326377 for the main diagonal of T.
%Y Cf. A006519, A061142, A297473, A297845, A319525.
%K nonn,tabl
%O 1,2
%A _Rémy Sigrist_, Jul 02 2019