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A059896 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. 25
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
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.
A003418-analog seems to be A066616. - Antti Karttunen, Apr 12 2017
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
LINKS
Eric Weisstein's World of Mathematics, Square Part.
Eric Weisstein's World of Mathematics, Squarefree Part.
FORMULA
From Antti Karttunen, Apr 11 2017: (Start)
A(x,y) = A059895(x,y) * A059897(x,y).
A(x,y) * A059895(x,y) = x*y.
(End).
From Peter Munn, Mar 02 2022: (Start)
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.
A(n, k) = A(A007913(n), A007913(k)) * A(A008833(n), A008833(k)) = lcm(A007913(n), A007913(k)) * A(A000188(n), A000188(k))^2.
A007947(A(n, k)) = A007947(n*k).
Isomorphism: A(A052330(n), A052330(k)) = A052330(n OR k).
Equivalently, A(n, k) = A052330(A052331(n) OR A052331(k)).
A(A003961(n), A003961(k)) = A003961(A(n, k)).
A(A225546(n), A225546(k)) = A225546(A(n, k)).
(End)
EXAMPLE
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
MATHEMATICA
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]]])];
Table[a[i - j + 1, j], {i, 1, 15}, {j, 1, i}] // Flatten (* Jean-François Alcover, Jun 19 2018 *)
PROG
(Scheme)
(define (A059896 n) (A059896bi (A002260 n) (A004736 n)))
(define (A059896bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) (* m b)) ((= 1 b) (* m a)) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A003986bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (/ a (A028233 a)) b (* m (A028233 a)))) (else (loop a (/ b (A028233 b)) (* m (A028233 b)))))))
;; Antti Karttunen, Apr 11 2017
(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
CROSSREFS
Sequences used in a definition of this sequence: A003986, A000188/A007913/A008833, A052330/A052331.
Has simple/very significant relationships with A003961, A059895/A059897, A225546, A267116.
Sequence in context: A003990 A287958 A308630 * A079542 A220370 A291048
KEYWORD
base,easy,nonn,tabl
AUTHOR
Marc LeBrun, Feb 06 2001
EXTENSIONS
New name from Peter Munn, Mar 02 2022
STATUS
approved

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)