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A059896
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The set of Fermi-Dirac factors of A(n,k) is the union of the Fermi-Dirac factors of n and k. Symmetric square array read by antidiagonals.
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25
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1, 2, 2, 3, 2, 3, 4, 6, 6, 4, 5, 8, 3, 8, 5, 6, 10, 12, 12, 10, 6, 7, 6, 15, 4, 15, 6, 7, 8, 14, 6, 20, 20, 6, 14, 8, 9, 8, 21, 24, 5, 24, 21, 8, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 10, 27, 8, 35, 6, 35, 8, 27, 10, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24
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OFFSET
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1,2
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COMMENTS
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Every positive integer, m, is the product of a unique subset, S(m), of the numbers listed in A050376 (primes, squares of primes etc.) The Fermi-Dirac factors of m are the members of S(m). So T(n,k) is the product of the members of (S(n) U S(k)).
Old name: Table a(i,j) = product prime(k)^(Ei(k) OR Ej(k)) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; OR is the bitwise operation on binary representation of the exponents.
Analogous to LCM, with OR replacing MAX.
Considered as a binary operation, the result is the lowest common multiple of the squarefree parts of its operands multiplied by the square of the operation's result when applied to the square roots of the square parts of its operands. - Peter Munn, Mar 02 2022
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LINKS
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FORMULA
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(End).
OR denotes the bitwise operation (A003986).
Limited multiplicative property: if gcd(n_1*k_1, n_2*k_2) = 1 then A(n_1*n_2, k_1*k_2) = A(n_1, k_1) * A(n_2, k_2).
For prime p, A(p^e_1, p^e_2) = p^(e_1 OR e_2).
A(n, A(m, k)) = A(A(n, m), k).
A(n, k) = A(k, n).
A(n, 1) = A(n, n) = n.
A(n^2, k^2) = A(n, k)^2.
(End)
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EXAMPLE
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A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 OR 3) * 3^(3 OR 5) = 2^7*3^7 = 279936.
The top left 12 X 12 corner of the array:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
2, 2, 6, 8, 10, 6, 14, 8, 18, 10, 22, 24
3, 6, 3, 12, 15, 6, 21, 24, 27, 30, 33, 12
4, 8, 12, 4, 20, 24, 28, 8, 36, 40, 44, 12
5, 10, 15, 20, 5, 30, 35, 40, 45, 10, 55, 60
6, 6, 6, 24, 30, 6, 42, 24, 54, 30, 66, 24
7, 14, 21, 28, 35, 42, 7, 56, 63, 70, 77, 84
8, 8, 24, 8, 40, 24, 56, 8, 72, 40, 88, 24
9, 18, 27, 36, 45, 54, 63, 72, 9, 90, 99, 108
10, 10, 30, 40, 10, 30, 70, 40, 90, 10, 110, 120
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 11, 132
12, 24, 12, 12, 60, 24, 84, 24, 108, 120, 132, 12
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MATHEMATICA
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a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, e1[_] = 0; Scan[(e1[#[[1]]] = #[[2]])&, f1]; e2[_] = 0; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitOr[e1[#], e2[#]]& /@ Union[f1[[All, 1]], f2[[All, 1]]])];
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PROG
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(Scheme)
(PARI) A059896(n, k) = if(n==k, n, lcm(core(n), core(k)) * A059896(core(n, 1)[2], core(k, 1)[2])^2) \\ Peter Munn, Mar 07 2022
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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