[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) ;
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