# [seqfan] Partitions of n into squared divisors

Richard Mathar mathar at strw.leidenuniv.nl
Sun May 10 18:42:10 CEST 2009

```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) ;

```