A326376 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).
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, 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, 2, 1, 4, 2, 10, 16, 8, 47250, 14, 512, 4, 10, 2, 1, 1, 8
Offset: 1
Examples
Array T(n, k) begins:
n\k| 1 2 3 4 5 6 7 8 9 10
---+-----------------------------------------------------------
1| 1 1 1 1 1 1 1 1 1 1
2| 2 2 2 2 2 2 2 2 2 2
3| 1 2 3 4 5 6 7 8 9 10
4| 4 4 4 4 4 4 4 4 4 4
5| 1 2 5 16 11 90 17 512 625 550
6| 2 4 6 8 10 12 14 16 18 20
7| 1 2 7 256 17 47250 29 134217728 5764801 5656750
8| 8 8 8 8 8 8 8 8 8 8
9| 1 4 9 16 25 36 49 64 81 100
10| 2 4 10 32 22 180 34 1024 1250 1100
The corresponding polynomials are:
f(n)\f(k)| 0 1 x 2 x^2 x+1 x^3 3 2*x x^2+1
---------+---------------------------------------------------------------------
0| 0 0 0 0 0 0 0 0 0 0
1| 1 1 1 1 1 1 1 1 1 1
x| 0 1 x 2 x^2 x+1 x^3 3 2*x x^2+1
2| 2 2 2 2 2 2 2 2 2 2
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
x+1| 1 2 x+1 3 x^2+1 x+2 x^3+1 4 2*x+1 x^2+2
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
3| 3 3 3 3 3 3 3 3 3 3
2*x| 0 2 2*x 4 2*x^2 2*x+2 2*x^3 6 4*x 2*x^2+2
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
Programs
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PARI
g(p) = my (c=Vecrev(Vec(p))); prod (i=1, #c, if (c[i], prime(i)^c[i], 1)) f(n, v='x) = my (f=factor(n)); sum (i=1, #f~, f[i,2] * v^(primepi(f[i,1]) - 1)) T(n,k) = g(f(n, f(k)))
Comments