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A326376
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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).
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2
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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
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OFFSET
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1,2
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COMMENTS
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This sequence has connections with A297845.
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.
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LINKS
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FORMULA
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For any m, n, k > 0 and any i >= 0:
- T(1, k) = 1,
- T(2^i, k) = 2^i,
- T(3, k) = k,
- T(3^i, k) = k^i,
- T(6, k) = 2*k,
- T(n, 3) = n,
- T(m*n, k) = T(m, k) * T(n, k).
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EXAMPLE
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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
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PROG
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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)))
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CROSSREFS
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See A326377 for the main diagonal of T.
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KEYWORD
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AUTHOR
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STATUS
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approved
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