A059897 Symmetric square array read by antidiagonals: A(n,k) is the product of all factors that occur in one, but not both, of the Fermi-Dirac factorizations of n and k.
1, 2, 2, 3, 1, 3, 4, 6, 6, 4, 5, 8, 1, 8, 5, 6, 10, 12, 12, 10, 6, 7, 3, 15, 1, 15, 3, 7, 8, 14, 2, 20, 20, 2, 14, 8, 9, 4, 21, 24, 1, 24, 21, 4, 9, 10, 18, 24, 28, 30, 30, 28, 24, 18, 10, 11, 5, 27, 2, 35, 1, 35, 2, 27, 5, 11, 12, 22, 30, 36, 40, 42, 42, 40, 36, 30, 22, 12, 13, 24, 33
Offset: 1
Examples
A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 XOR 3) * 3^(3 XOR 5) = 2^6 * 3^6 = 46656.
The top left 12 X 12 corner of the array:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
2, 1, 6, 8, 10, 3, 14, 4, 18, 5, 22, 24
3, 6, 1, 12, 15, 2, 21, 24, 27, 30, 33, 4
4, 8, 12, 1, 20, 24, 28, 2, 36, 40, 44, 3
5, 10, 15, 20, 1, 30, 35, 40, 45, 2, 55, 60
6, 3, 2, 24, 30, 1, 42, 12, 54, 15, 66, 8
7, 14, 21, 28, 35, 42, 1, 56, 63, 70, 77, 84
8, 4, 24, 2, 40, 12, 56, 1, 72, 20, 88, 6
9, 18, 27, 36, 45, 54, 63, 72, 1, 90, 99, 108
10, 5, 30, 40, 2, 15, 70, 20, 90, 1, 110, 120
11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 1, 132
12, 24, 4, 3, 60, 8, 84, 6, 108, 120, 132, 1
From _Peter Munn_, Apr 04 2019: (Start)
The subgroup generated by {6,8,10}, the first three integers > 1 not in A050376, has the following table:
1 6 8 10 12 15 20 120
6 1 12 15 8 10 120 20
8 12 1 20 6 120 10 15
10 15 20 1 120 6 8 12
12 8 6 120 1 20 15 10
15 10 120 6 20 1 12 8
20 120 10 8 15 12 1 6
120 20 15 12 10 8 6 1
(End)
Links
- Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array
- Eric Weisstein's World of Mathematics, Group, Square Part, Squarefree Part.
Crossrefs
Cf. A000040, A003987, A003991, A028233, A028234, A050376, A059896, A089913, A207901, A268387, A284577, A302033.
Rows/columns: A073675 (2), A120229 (3), A120230 (4), A307151 (5), A307150 (6), A307266 (8), A307267 (24).
Programs
-
Mathematica
a[i_, i_] = 1; 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 @@ (#^BitXor[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 *) -
PARI
T(n,k) = {if (n==1, return (k)); if (k==1, return (n)); my(fn=factor(n), fk=factor(k)); vp = setunion(fn[,1]~, fk[,1]~); prod(i=1, #vp, vp[i]^(bitxor(valuation(n, vp[i]), valuation(k, vp[i]))));} \\ Michel Marcus, Apr 03 2019 -
PARI
T(i, j) = {if(gcd(i, j) == 1, return(i * j)); if(i == j, return(1)); my(f = vecsort(concat(factor(i)~, factor(j)~)), t = 1, res = 1); while(t + 1 <= #f, if(f[1, t] == f[1, t+1], res *= f[1, t] ^ bitxor(f[2, t] , f[2, t+1]); t+=2; , res*= f[1, t]^f[2, t]; t++; ) ); if(t == #f, res *= f[1, #f] ^ f[2, #f]); res } \\ David A. Corneth, Apr 03 2019 -
PARI
A059897(n,k) = if(n==k, 1, core(n*k) * A059897(core(n,1)[2],core(k,1)[2])^2) \\ Peter Munn, Mar 21 2022
-
Scheme
(define (A059897 n) (A059897bi (A002260 n) (A004736 n))) (define (A059897bi 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) (A003987bi (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
Formula
For all x, y >= 1, A(x,y) * A059895(x,y)^2 = x*y. - Antti Karttunen, Apr 11 2017
From Peter Munn, Apr 01 2019: (Start)
A(n,1) = A(1,n) = n
A(n, A(m,k)) = A(A(n,m), k)
A(n,n) = 1
A(n,k) = A(k,n)
if A(n,k_1) = n * k_1 and A(n,k_2) = n * k_2 then A(n, A(k_1,k_2)) = n * A(k_1,k_2)
(End)
T(k, m) = k*m for coprime k and m. - David A. Corneth, Apr 03 2019
if A(n*m,m) = n, A(n*m,k) = A(n,k) * A(m,k) / k. - Peter Munn, Apr 04 2019
Extensions
New name from Peter Munn, Mar 21 2022
Comments