A090677 Number of ways to partition n into sums of squares of primes.

1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 3, 2, 2, 2, 4, 3, 2, 3, 4, 4, 2, 3, 4, 5, 3, 3, 5, 5, 4, 3, 5, 5, 5, 4, 5, 6, 5, 5, 5, 7, 6, 6, 6, 7, 7, 6, 7, 7, 8, 7, 8, 8, 8, 8, 8, 9, 8, 9, 9, 10, 9, 9, 10, 11, 11, 10, 11

Offset: 0

Author

N. J. A. Sloane, Dec 19 2003

Keywords

• nonn

Comments

From Hieronymus Fischer, Nov 11 2007: (Start)

First statement of monotony: a(n+p^2)>=a(n) for all primes p. Proof: we restrict ourselves on a(n)>0 (the case a(n)=0 is trivial). Let T(i), 1<=i<=a(n), be the a(n) different sums of squares of primes representing n. Then, adding p^2 to those expressions, we get a(n) sums of squares of primes T(i)+p^2, obviously representing n+p^2, thus a(n+p^2) cannot be less than a(n).

Second statement of monotony: a(n+m)>=max(a(n),a(m)) for all m with a(m)>1. Proof: let T(i), 1<=i<=a(n), be the a(n) different sums of squares of primes representing n; let S(i), 1<=i<=a(m), be the a(m) different sums of squares of primes representing m. Then, adding these expressions, we get a(n) sums of squares of primes T(i)+S(1), representing n+m, further we get a(m) sums T(1)+S(i), also representing n+m. Thus a(n+m) cannot be less than the maximum of a(n) and a(m).

The minimum b(k):=min( n | a(j)>k for all j>n) exists for all k>=0. See A134755 for that sequence representing b(k). (End)

Examples

			a(25)=2 because 25 = 5^2 = 4*(2^2)+3^2.
a(83)=8 because 83 = 3^2+5^2+7^2 = 4*(2^2)+2*(3^2)+7^2
                   = 2*(2^2)+3*(5^2) = 6*(2^2)+3^2+2*(5^2)
                   = 2^2+6*(3^2)+5^2 = 10*(2^2)+2*(3^2)+5^2
                   = 5*(2^2)+7*(3^2) = 14*(2^2)+3*(3^2).
		

References

• R. F. Churchouse, Representation of integers as sums of squares of primes. Caribbean J. Math. 5 (1986), no. 2, 59-65.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Christian N. K. Anderson, Table of coefficients b_i solving n=sum(b_i * prime(i)^2), for n=0..500
• Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.

Crossrefs

Cf. A111900, A001248, A001156, A023893, A111901.

Cf. A078134, A078135, A078136, A078139, A134600, A078137, A134754, A134755.

Programs

Mathematica

CoefficientList[ Series[ Product[1/(1 - x^Prime[i]^2), {i, 111}], {x, 0, 101}], x] (* Robert G. Wilson v, Sep 20 2004 *)

Formula

G.f.: 1/((1-x^4)*(1-x^9)*(1-x^25)*(1-x^49)*(1-x^121)*(1-x^169)*(1-x^289)...).

G.f.: 1 + Sum_{i>=1} x^(prime(i)^2) / Product_{j=1..i} (1 - x^(prime(j)^2)). - Ilya Gutkovskiy, May 07 2017