A062831 Number of ways n can be expressed as the sum of a nonzero square and 1 or a prime.

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

Offset

1,6

Links

Table of n, a(n) for n=1..102.

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.

Mathematica

a[n_] := Length[Select[n-Range[1, Floor[Sqrt[n]]]^2, #==1||PrimeQ[ # ]&]]

Crossrefs

Sequence in context: A157047 A059342 A357481 * A037828 A030419 A155052

Adjacent sequences: A062828 A062829 A062830 * A062832 A062833 A062834

Keyword

nonn

Author

Santi Spadaro, Jul 20 2001

Extensions

Corrected and extended by Dean Hickerson, Jul 26, 2001

Status

approved