|
| |
|
|
A090677
|
|
Number of ways to partition n into sums of squares of primes.
|
|
10
| |
|
|
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
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,26
|
|
|
COMMENTS
| 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)
|
|
|
REFERENCES
| 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.
|
|
|
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)...).
|
|
|
MATHEMATICA
| CoefficientList[ Series[ Product[1/(1 - x^Prime[i]^2), {i, 111}], {x, 0, 101}], x] (from 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: A136177 A066922 A033183 * A161097 A105240 A143654
Adjacent sequences: A090674 A090675 A090676 * A090678 A090679 A090680
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Dec 19 2003
|
| |
|
|
