OFFSET
0,2
COMMENTS
Consider the g.f. of the prime parts partition numbers, GF=1/product(1-x^prime(k),k=1,...), cf A000607. Then consecutively subtract c(n)*x^a(n)/product(1-x^k,k=1..n), n=0,1,2,3,... where c(n)*x^a(n) is the leading term of the remaining expression (GF - previously subtracted terms).
The identity is wrong since the pattern of a(n)=sum of first n primes (cf. A046676) breaks after a(4)=17; the pattern c(n)=1 breaks also after n=4, and the nontrivial cancellations stop after the power a(5)=21, followed by 22, 24, 25, 26, 27, ...
EXAMPLE
GF = 1/((1-x^2)(1-x^3)(1-x^5)(1-x^7)(1-x^11)...) = 1+x^2+x^3+x^4+2*x^5+2*x^6+... (cf. A000607)
a(0)=0, c(0)=1: GF - 1 = x^2 + ....
a(1)=2, c(1)=1: GF - 1 - x^2/(1-x) = x^5 + ...
a(2)=5, c(2)=1: GF - 1 - x^2/(1-x) - x^5/(1-x)(1-x^2) = x^10 + ...
a(3)=10, c(3)=1: GF - ... - x^10/(1-x)(1-x^2)(1-x^3) = x^17 + ...
a(4)=17, c(4)=1: GF - ... - x^17/(1-x)(1-x^2)(1-x^3)(1-x^4) = -x^21 + ...
a(5)=21, c(5)=-1: GF - ... + x^21/... etc.
PROG
(PARI) p=1/prod(k=1, 25, 1-x^prime(k), 1+O(x^99)); for(k=0, 9, [print1(c=valuation(p, x), ", "), c=polcoeff(p, c)*x^c/prod(j=1, k, 1-x^j), "\n", p-=c])
CROSSREFS
KEYWORD
nonn
AUTHOR
M. F. Hasler, Mar 05 2014
EXTENSIONS
Example section corrected by Vaclav Kotesovec, Sep 12 2019
STATUS
approved