A238804 - OEIS

%I #12 Sep 12 2019 04:42:26

%S 0,2,5,10,17,21,22,24,25,26,27,28,29,30,31,32,33,34,36,37,39,41,44,45,

%T 46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,

%U 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

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

%C 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).

%C 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, ...

%e 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)

%e a(0)=0, c(0)=1: GF - 1 = x^2 + ....

%e a(1)=2, c(1)=1: GF - 1 - x^2/(1-x) = x^5 + ...

%e a(2)=5, c(2)=1: GF - 1 - x^2/(1-x) - x^5/(1-x)(1-x^2) = x^10 + ...

%e a(3)=10, c(3)=1: GF - ... - x^10/(1-x)(1-x^2)(1-x^3) = x^17 + ...

%e a(4)=17, c(4)=1: GF - ... - x^17/(1-x)(1-x^2)(1-x^3)(1-x^4) = -x^21 + ...

%e a(5)=21, c(5)=-1: GF - ... + x^21/... etc.

%o (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])

%K nonn

%O 0,2

%A _M. F. Hasler_, Mar 05 2014

%E Example section corrected by _Vaclav Kotesovec_, Sep 12 2019