# [seqfan] a simple Partitions question

wouter meeussen wouter.meeussen at pandora.be
Sat Mar 20 19:54:18 CET 2010

```When looking at strict partitions of n into positive powers of 3, 5 and 7,
I noticed a few surprises (at least to me).
First, the list of integers that cannot be so partitioned starts off quite
dense, as
1 2 4 6 11 13 18 20 22 23 26 29 31 38 45 47 50 53 72 75 78 80 87 94 99 103
107 112 ...
but then the count of such partitions, say P357(n), takes aflight and shows
surprising symmetry:
P357(n) = P357(2270   -n) for n> 84
P357(n) = P357(16545 -n) for n>920
P357(n) = P357(68660 -n) for n>9611
(Is this caused by some Modular hocus-pocus?).
I browsed in vain for 2270, 16545, 68660
It is also striking that P357(n) keeps to fairly low values,
reaching a local max of just 55 at n=26399 and 42261 and only A143838 is
close.
Could anyone show me where the symmetry comes from?

Wouter.
----------------------------------------------------------
for my records:
n=68660;Timing[P357=Rest at CoefficientList[ Series[
Product[1+x^(3^k),{k,Ceiling[Log[3,n]]}]*
Product[1+x^(5^k),{k,Ceiling[Log[5,n]]}]*
Product[1+x^(7^k),{k,Ceiling[Log[7,n]]}],{x,0,n}],x];]
-----------------------------------------------------------

```