0,26

Comments from Hieronymus Fischer (Hieronymus.Fischer(AT)gmx.de), 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)

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

Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.

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

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

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

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

Sequence in context: A136177 A066922 A033183 this_sequence A161097 A105240 A143654

Adjacent sequences: A090674 A090675 A090676 this_sequence A090678 A090679 A090680

nonn

N. J. A. Sloane (njas(AT)research.att.com), Dec 19 2003