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.

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

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)
    h_values = ((x, h(x)) for x in range(1, n + 1))
    return any(hx != 0 and hx % x == 0 and hx == x*k(x) for x, hx in h_values)
[n for n in range(267) if is_A197099(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