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

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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 7, 2, 1

Offset: 1

Author

Amiram Eldar, Dec 11 2018

Keywords

• nonn
• tabl

Comments

This relation was defined by Cohen in 1960.

The common notation for T(n,k) is (n,k)*.

If T(n,k) = 1 then n is said to be semi-prime to k.

In general T(n,k) != T(k,n).

The relation is used to define semi-unitary divisors (A322483).

Examples

			The table starts
  1  1  1  1  1  1  1  1  1  1 ...
  1  2  1  1  1  2  1  1  1  2 ...
  1  1  3  1  1  3  1  1  1  1 ...
  1  2  1  4  1  2  1  1  1  2 ...
  1  1  1  1  5  1  1  1  1  5 ...
  1  2  3  1  1  6  1  1  1  2 ...
  1  1  1  1  1  1  7  1  1  1 ...
  1  2  1  4  1  2  1  8  1  2 ...
  1  1  3  1  1  3  1  1  9  1 ...
  1  2  1  1  5  2  1  1  1 10 ...
  ...
The triangle formed by the antidiagonals starts
  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 1 1
  1 1 1 2 5 1 1 2 1
  ...
		

References

• J. Sandor and B. Crstici, Handbook of Number Theory, II, Springer Verlag, 2004, chapter 3.6, pp. 281.

Links

• Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Mathematische Zeitschrift, Vol. 74, No. 1 (1960), pp. 66-80.
• M. V. Subbarao On some arithmetic convolutions, The theory of arithmetic functions, Springer, Berlin, Heidelberg, 1972, pp. 247-271.
• D. Suryanarayana and V. Siva Rama Prasad, Sum functions of k-ary and semi-k-ary divisors, Journal of the Australian Mathematical Society, Vol. 15, No. 2 (1973), pp. 148-162.

Crossrefs

Cf. A050873 (gcd), A165430 (unitary gcd).

Programs

Mathematica

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

PARI

udivisors(n) = {my(d=divisors(n)); select(x->(gcd(x, n/x)==1), d);}
T(n,k) = {my(dn = divisors(n), udk = udivisors(k)); vecmax(setintersect(dn, udk));} \\ Michel Marcus, Dec 14 2018

Formula

T(1,n) = T(n,1) = 1.

T(n,n) = n.