[seqfan] Re: Partitions of n into squared divisors

[franktaw at netscape.net]

Using the following PARI program:

a(n)=local(d);d=divisors(n);polcoeff(prod(i=1,#d,1/(1-x^d[i]^2+x*O(x^n)))
,n)

I get:

1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 2, 6, 1, 9, 1, 8, 3, 6, 1, 
16, 2, 7, 4, 12, 1, 21, 1, 15, 4, 9, 2, 39, 1, 10, 5, 25, 1, 35, 1, 24, 
9, 12, 1, 76, 2, 21, 6, 32, 1, 61, 3, 38, 7, 15, 1, 174, 1, 16, 10, 46, 
3, 93, 1, 50, 8, 42, 1, 231, 1, 19, 19, 60, 2, 135, 1, 118

This agrees with your values, and is definitely not in the database.

Franklin T. Adams-Watters


-----Original Message-----
From: Richard Mathar <mathar at strw.leidenuniv.nl>

Is the number of partitions of n, such that each part is a square of a
divisor of n, in the OEIS? This is related to A018818, which demands 
that
each part is a divisor of n, and is related to the question in how many 
ways
a group of order n allows decomposition into irreducible subgroups 
where the
characters need to be divisors of the group order. Maybe I am 
overlooking
something.
This sequence ought start at n=1 (ie, with offset 1) as

       1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 5, 1, 4, 2, 6, 1, 9, 1, 8, 3, 6, 
1
and represent the following partitions n [terms of partition] for small 
n:

                                     1, [1]

                                   2, [1, 1]

                                  3, [1, 1, 1]

                                4, [1, 1, 1, 1]

                                     4, [4]

                               5, [1, 1, 1, 1, 1]

                             6, [1, 1, 1, 1, 1, 1]

                                  6, [1, 1, 4]

                            7, [1, 1, 1, 1, 1, 1, 1]

                          8, [1, 1, 1, 1, 1, 1, 1, 1]

                               8, [1, 1, 1, 1, 4]

                                   8, [4, 4]

                         9, [1, 1, 1, 1, 1, 1, 1, 1, 1]

                                     9, [9]

                       10, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

                           10, [1, 1, 1, 1, 1, 1, 4]

                                10, [1, 1, 4, 4]

                     11, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

                    12, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

                        12, [1, 1, 1, 1, 1, 1, 1, 1, 4]

                             12, [1, 1, 1, 1, 4, 4]

                                12, [1, 1, 1, 9]

                                 12, [4, 4, 4]

                  13, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

                 14, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

                     14, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4]

                          14, [1, 1, 1, 1, 1, 1, 4, 4]

                              14, [1, 1, 4, 4, 4]

               15, [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]

                           15, [1, 1, 1, 1, 1, 1, 9]

Recursive Maple program:
Nrep := proc(n,minEl,setd)
    local a,d ;
        a := 0 ;
        for d in setd do
                if d >= minEl then
                        if d^2 = n then
                                a := a+1 ;
                        elif d > n then
                                ;
                        else
                                a := a+ Nrep(n-d^2,d,setd) ;
                        fi;
                fi;
        od:
        a ;
end:

nrepsq := proc(n)
        Nrep(n,1,numtheory[divisors](n) ) ;
end:

seq(nrepsq(n),n=1..23) ;



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