A059895 Table a(i,j) = product prime[k]^(Ei[k] AND Ej[k]) where Ei and Ej are the vectors of exponents in the prime factorizations of i and j; AND is the bitwise operation on binary representation of the exponents.

1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 2, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 2, 1, 1, 5, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 1, 6, 1, 4, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 7, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 5, 1, 1, 1, 1, 5, 1, 3, 1, 1, 1, 2, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 2, 1

Offset

1,5

Comments

Analogous to GCD, with AND replacing MIN.

Links

Antti Karttunen, Table of n, a(n) for n = 1..10440; the first 144 antidiagonals of the array

Formula

From Antti Karttunen, Apr 11 2017: (Start)

A(x,y) = A059896(x,y) / A059897(x,y).

A(x,y) * A059896(x,y) = A(x,y)^2 * A059897(x,y) = x*y.

(End)

Example

The top left 18 X 18 corner of the array:
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  1,  2,  1,  2,  1,  1,  1,  2,  1,  1,  1,  2
1,  1,  3,  1,  1,  3,  1,  1,  1,  1,  1,  3,  1,  1,  3,  1,  1,  1
1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  4,  1,  1,  1,  1,  1,  1
1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1,  1,  5,  1,  1,  1
1,  2,  3,  1,  1,  6,  1,  2,  1,  2,  1,  3,  1,  2,  3,  1,  1,  2
1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1,  1,  1,  7,  1,  1,  1,  1
1,  2,  1,  4,  1,  2,  1,  8,  1,  2,  1,  4,  1,  2,  1,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  9,  1,  1,  1,  1,  1,  1,  1,  1,  9
1,  2,  1,  1,  5,  2,  1,  2,  1, 10,  1,  1,  1,  2,  5,  1,  1,  2
1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 11,  1,  1,  1,  1,  1,  1,  1
1,  1,  3,  4,  1,  3,  1,  4,  1,  1,  1, 12,  1,  1,  3,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 13,  1,  1,  1,  1,  1
1,  2,  1,  1,  1,  2,  7,  2,  1,  2,  1,  1,  1, 14,  1,  1,  1,  2
1,  1,  3,  1,  5,  3,  1,  1,  1,  5,  1,  3,  1,  1, 15,  1,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 16,  1,  1
1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, 17,  1
1,  2,  1,  1,  1,  2,  1,  2,  9,  2,  1,  1,  1,  2,  1,  1,  1, 18
A(864,1944) = A(2^5*3^3,2^3*3^5) = 2^(5 AND 3)* 3^(3 AND 5) = 2^1*3^1 = 6.

Mathematica

a[i_, i_] := i;
a[i_, j_] := Module[{f1 = FactorInteger[i], f2 = FactorInteger[j], e1, e2}, Scan[(e1[#[[1]]] = #[[2]])&, f1]; Scan[(e2[#[[1]]] = #[[2]])&, f2]; Times @@ (#^BitAnd[e1[#], e2[#]]& /@ Intersection[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 (A059895 n) (A059895bi (A002260 n) (A004736 n)))
(define (A059895bi a b) (let loop ((a a) (b b) (m 1)) (cond ((= 1 a) m) ((= 1 b) m) ((equal? (A020639 a) (A020639 b)) (loop (A028234 a) (A028234 b) (* m (expt (A020639 a) (A004198bi (A067029 a) (A067029 b)))))) ((< (A020639 a) (A020639 b)) (loop (A028234 a) b m)) (else (loop a (A028234 b) m)))))
;; Antti Karttunen, Apr 11 2017

Crossrefs

Cf. A003985 (A004198), A003989, A028234, A059896, A059897, A067029, A267115, A284578.

Sequence in context: A204125 A204127 A225174 * A368328 A321167 A190867

Adjacent sequences: A059892 A059893 A059894 * A059896 A059897 A059898

Keyword

base,easy,nonn,tabl

Author

Marc LeBrun, Feb 06 2001

Extensions

Data section extended to 120 terms by Antti Karttunen, Apr 11 2017

Status

approved