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

1,5

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).

References

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

Links

Table of n, a(n) for n=1..87.

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.

Formula

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

T(n,n) = n.

Example

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
  ...

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

Prog

(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

Crossrefs

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

Sequence in context: A204112 A186027 A359250 * A231071 A209156 A329325

Adjacent sequences: A322479 A322480 A322481 * A322483 A322484 A322485

Keyword

nonn,tabl

Author

Amiram Eldar, Dec 11 2018

Status

approved