

A090677


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


18



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
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OFFSET

0,26


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)


REFERENCES

R. F. Churchouse, Representation of integers as sums of squares of primes. Caribbean J. Math. 5 (1986), no. 2, 5965.


LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Roger Woodford, Bounds for the Eventual Positivity of Difference Functions of Partitions, Journal of Integer Sequences, Vol. 10 (2007), Article 07.1.3.


FORMULA

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


MATHEMATICA

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


CROSSREFS

Cf. A111900, A001248, A001156, A023893, A111901.
Cf. A078134, A078135, A078136, A078139, A134600, A078137, A134754, A134755.
Sequence in context: A212178 A066922 A033183 * A161097 A105240 A227481
Adjacent sequences: A090674 A090675 A090676 * A090678 A090679 A090680


KEYWORD

nonn


AUTHOR

N. J. A. Sloane, Dec 19 2003


STATUS

approved



