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A197099 Define the array k(n,x) = number of m such that tau(gcd(n,m)) is x where m runs from 1 to n. Also define h(n,x) = Sum_{d|n : tau(d) = x} d. The sequence contains numbers n such that k(n,x)*x = h(n,x) has at least one solution x. 0
1, 2, 4, 32, 48, 180, 189, 224, 288, 360, 432, 1280, 1344, 1536, 1600, 4096, 28672, 46656, 54000, 108000, 131220, 150528, 225792, 262440, 405450, 442800, 525312, 532480, 590400, 594000, 630784, 633600, 655360, 792000, 819200, 885600, 950400 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

In the definition tau=A000005. By construction of the two arrays, their row sums and/or first moments are Sum_{x=1..z} k(x)*x = Sum_{x=1..z} h(x) = sigma(n) = A000203(n).

From R. J. Mathar, Oct 12 2011: (Start)

The table k(n,x) with row sums n is a frequency distribution of tau which starts in row n=1 with columns x >= 1 as follows:

   1,  0,  0,  0,  0,  0,  0,  0, ...

   1,  1,  0,  0,  0,  0,  0,  0, ...

   2,  1,  0,  0,  0,  0,  0,  0, ...

   2,  1,  1,  0,  0,  0,  0,  0, ...

   4,  1,  0,  0,  0,  0,  0,  0, ...

   2,  3,  0,  1,  0,  0,  0,  0, ...

   6,  1,  0,  0,  0,  0,  0,  0, ...

   4,  2,  1,  1,  0,  0,  0,  0, ...

   6,  2,  1,  0,  0,  0,  0,  0, ...

   4,  5,  0,  1,  0,  0,  0,  0, ...

  10,  1,  0,  0,  0,  0,  0,  0, ...

   4,  4,  2,  1,  0,  1,  0,  0, ...

By multiplying with the column number x we obtain another array x*k(n,x) which has row sums sigma(n):

   1,  0,  0,  0,  0,  0,  0,  0, ...

   1,  2,  0,  0,  0,  0,  0,  0, ...

   2,  2,  0,  0,  0,  0,  0,  0, ...

   2,  2,  3,  0,  0,  0,  0,  0. ...

   4,  2,  0,  0,  0,  0,  0,  0, ...

   2,  6,  0,  4,  0,  0,  0,  0, ...

   6,  2,  0,  0,  0,  0,  0,  0, ...

   4,  4,  3,  4,  0,  0,  0,  0, ...

   6,  4,  3,  0,  0,  0,  0,  0, ...

   4, 10,  0,  4,  0,  0,  0,  0, ...

  10,  2,  0,  0,  0,  0,  0,  0, ...

   4,  8,  6,  4,  0,  6,  0,  0, ...

The array h(n,x) with another frequency distribution of tau and also rows sums sigma(n) starts in row n=1 as follows:

   1,  0,  0,  0,  0,  0,  0,  0, ...

   1,  2,  0,  0,  0,  0,  0,  0, ...

   1,  3,  0,  0,  0,  0,  0,  0, ...

   1,  2,  4,  0,  0,  0,  0,  0, ...

   1,  5,  0,  0,  0,  0,  0,  0, ...

   1,  5,  0,  6,  0,  0,  0,  0, ...

   1,  7,  0,  0,  0,  0,  0,  0, ...

   1,  2,  4,  8,  0,  0,  0,  0, ...

   1,  3,  9,  0,  0,  0,  0,  0, ...

   1,  7,  0, 10,  0,  0,  0,  0, ...

   1, 11,  0,  0,  0,  0,  0,  0, ...

   1,  5,  4,  6,  0, 12,  0,  0, ...

Whenever the previous two tables match at one position (n,x) for a nonzero entry, we add the corresponding row number n to the sequence. The rows at n=4, (2,2,3) and (1,2,4) for example, match at x=2, which adds n=4 to the sequence. (End)

LINKS

Table of n, a(n) for n=1..37.

EXAMPLE

For n = 189: 21|189, 27|189 and tau(21) = tau(27) = 4; h(4) = Sum_{d|189; tau(d) = 4} d = 21+27 = k(4)*4 = 12*4 = 48. Therefore 189 is in the sequence.

MAPLE

k := proc(n, x)

        a := 0 ;

        for m from 1 to n do

                if numtheory[tau](igcd(n, m)) = x then

                        a := a+1 ;

                end if;

        end do;

        a ;

end proc:

h := proc(n, x)

        a := 0 ;

        for d in numtheory[divisors](n) do

                if numtheory[tau](d) = x then

                        a := a+d ;

                end if;

        end do;

        a ;

end proc:

isA197099 := proc(n)

        for x from 1 to n do

                if h(n, x) = x*k(n, x) and h(n, x) <> 0 then

                        return true;

                end if;

        end do:

        false;

end proc:

for n from 1 do

        if isA197099(n) then

                print(n);

        end if;

end do: # R. J. Mathar, Oct 12 2011

PROG

(Sage)

def is_A197099(n): # extremely inefficient but useful for reference purposes

....k = lambda x: sum(1 for m in (1..n) if number_of_divisors(gcd(n, m))==x)

....h = lambda x: sum(d for d in divisors(n) if number_of_divisors(d)==x)

....return any(h(x) != 0 and h(x) % x == 0 and h(x) == x*k(x) for x in (1..n)) # D. S. McNeil, Oct 12 2011

CROSSREFS

Sequence in context: A299783 A293760 A101575 * A009098 A192387 A320624

Adjacent sequences:  A197096 A197097 A197098 * A197100 A197101 A197102

KEYWORD

nonn

AUTHOR

Naohiro Nomoto, Oct 10 2011

STATUS

approved

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Last modified October 18 23:39 EDT 2019. Contains 328211 sequences. (Running on oeis4.)