# number of partitions of n into a non-zero square and a prime or 1

Wed Sep 5 18:58:02 CEST 2001

```Someone can say something not trivial about the following?

A062831
Sequence:  0,1,1,1,1,2,1,1,1,1,2,2,0,2,1,1,2,2,1,2,2,1,2,1,0,2,3,2,1,2,
0,3,2,0,2,1,1,4,2,1,2,2,1,2,2,1,3,2,1,2,2,2,2,3,1,3,2,0,2,2,
0,4,2,0,3,3,2,4,2,1,2,3,1,1,3,1,4,2,1,3,1,2,5,3,0,3,3,2,2,2,
0,4,2,1,3,2,1,4,1,1,3,3
Name:      Number of ways n can be expressed as the sum of a nonzero
square and 1 or a
prime.
Formula:   Note that a(k^2)=0 or 1 since each prime can be written
only in one way as a
difference of squares: (n+b)^2-n^2=p where p is a prime, only if
b^2+2nb=b(b+2n) is prime, only if b=1. In that case
p=2n+1; since every
prime is an odd number we get an 1 in the distribution
of a(k^2) for each
odd number which is prime.
Mma:       a[n_]:=Length[Select[n-Range[1,Floor[Sqrt[n]]]^2,#==1||PrimeQ[
# ]&]]
Keywords:  nonn
Offset:    1
Author(s): Santi Spadaro (spados at katamail.com), Jul 20 2001
Extension: Corrected and extended by Dean Hickerson, Jul 26, 2001

Note that if we substitute "non-zero square" with "non-zero
triangular number", it is easy to show that a[T(n)]=0 or 1 following
the same pattern

regards,