A238804 Powers in a variant of Ramanujan's wrong identity for prime number partitions.

0, 2, 5, 10, 17, 21, 22, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 37, 39, 41, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99

Offset: 0

Author

M. F. Hasler, Mar 05 2014

Keywords

• nonn

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, ...

Examples

			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.
		

Programs

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])

Extensions

Example section corrected by Vaclav Kotesovec, Sep 12 2019