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A322482 Table read by downward antidiagonals: T(n,k) is the greatest divisor of n which is a unitary divisor of k.

%I A322482 #12 Dec 18 2018 11:29:06
%S A322482 1,1,1,1,2,1,1,1,1,1,1,1,3,2,1,1,1,1,1,1,1,1,2,1,4,1,2,1,1,1,3,1,1,3,
%T A322482 1,1,1,1,1,2,5,1,1,2,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,6,1,4,3,2,1,1,1,
%U A322482 1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,7,2,1
%N A322482 Table read by downward antidiagonals: T(n,k) is the greatest divisor of n which is a unitary divisor of k.
%C A322482 This relation was defined by Cohen in 1960.
%C A322482 The common notation for T(n,k) is (n,k)*.
%C A322482 If T(n,k) = 1 then n is said to be semi-prime to k.
%C A322482 In general T(n,k) != T(k,n).
%C A322482 The relation is used to define semi-unitary divisors (A322483).
%D A322482 J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 3.6, pp. 281.
%H A322482 Eckford Cohen, <a href="https://doi.org/10.1007/BF01180473">Arithmetical functions associated with the unitary divisors of an integer</a>, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
%H A322482 M. V. Subbarao <a href="https://doi.org/10.1007/BFb0058796">On some arithmetic convolutions</a>, The theory of arithmetic functions, Springer, Berlin, Heidelberg, 1972, pp. 247-271.
%H A322482 D. Suryanarayana and V. Siva Rama Prasad, <a href="https://doi.org/10.1017/S1446788700012908">Sum functions of k-ary and semi-k-ary divisors</a>, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.
%F A322482 T(1,n) = T(n,1) = 1.
%F A322482 T(n,n) = n.
%e A322482 The table starts
%e A322482   1  1  1  1  1  1  1  1  1  1 ...
%e A322482   1  2  1  1  1  2  1  1  1  2 ...
%e A322482   1  1  3  1  1  3  1  1  1  1 ...
%e A322482   1  2  1  4  1  2  1  1  1  2 ...
%e A322482   1  1  1  1  5  1  1  1  1  5 ...
%e A322482   1  2  3  1  1  6  1  1  1  2 ...
%e A322482   1  1  1  1  1  1  7  1  1  1 ...
%e A322482   1  2  1  4  1  2  1  8  1  2 ...
%e A322482   1  1  3  1  1  3  1  1  9  1 ...
%e A322482   1  2  1  1  5  2  1  1  1 10 ...
%e A322482   ...
%e A322482 The triangle formed by the antidiagonals starts
%e A322482   1
%e A322482   1 1
%e A322482   1 2 1
%e A322482   1 1 1 1
%e A322482   1 1 3 2 1
%e A322482   1 1 1 1 1 1
%e A322482   1 2 1 4 1 2 1
%e A322482   1 1 3 1 1 3 1 1
%e A322482   1 1 1 2 5 1 1 2 1
%e A322482   ...
%t A322482 udiv[n_] := Select[Divisors[n], GCD[#,n/#] == 1 &]; semiuGCD[a_, b_] := Max[ Intersection[Divisors[a], udiv[b]]]; Table[semiuGCD[n, k], {n,1,20}, {k, n-1, 1, -1 }] // Flatten
%o A322482 (PARI) udivisors(n) = {my(d=divisors(n)); select(x->(gcd(x, n/x)==1), d);}
%o A322482 T(n,k) = {my(dn = divisors(n), udk = udivisors(k)); vecmax(setintersect(dn, udk));} \\ _Michel Marcus_, Dec 14 2018
%Y A322482 Cf. A050873 (gcd), A165430 (unitary gcd).
%K A322482 nonn,tabl
%O A322482 1,5
%A A322482 _Amiram Eldar_, Dec 11 2018