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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. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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: A205617 A204112 A186027 * A231071 A209156 A191004

Adjacent sequences:  A322479 A322480 A322481 * A322483 A322484 A322485

KEYWORD

nonn,tabl

AUTHOR

Amiram Eldar, Dec 11 2018

STATUS

approved

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Last modified August 20 10:17 EDT 2019. Contains 326149 sequences. (Running on oeis4.)