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%I #16 Sep 12 2019 04:43:02
%S 1,1,1,1,1,-1,1,1,-1,1,-1,-1,1,-1,1,-2,2,-1,3,1,2,1,-1,-4,1,-4,-4,-10,
%T -2,-8,-4,-5,-4,-1,1,2,5,6,13,12,16,18,21,25,23,30,22,23,21,21,18,14,
%U 8,-1,-9,-20,-36,-36,-51,-61,-75,-80,-96,-103
%N Coefficients 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_{k>=1} (1-x^prime(k)), cf. A000607. Then consecutively subtract a(n)*x^b(n)/Product_{k=1..n} (1-x^k), n=0,1,2,3,... where a(n)*x^b(n) is the leading term of the remaining expression, GF - previously subtracted terms. Sequence A238804 lists the exponents b(n), here we list the coefficients a(n).
%C The identity Ramanujan considered, GF = Sum_{n>=0} x^Sum_{k=1..n} prime(k)/Product_{k=1..n} (1-x^k), or A000607 = A046676, is wrong: In the way they are defined above, the pattern of b(n) = (sum of first n primes) breaks after b(4)=17; the pattern a(n)=1 breaks also after n=4 (which yields this sequence), and the nontrivial cancellations stop after the power b(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+... (cf. A000607)
%e => a(0)=1, b(0)=0, GF - 1 = x^2 + ....
%e => a(1)=1, b(1)=2, GF - 1 - x^2/(1-x) = x^5 + ...
%e => a(2)=1, b(2)=5, GF - 1 - x^2/(1-x) - x^5/(1-x)(1-x^2) = x^10 + ...
%e => a(3)=1, b(3)=10, GF - ... - x^10/(1-x)(1-x^2)(1-x^3) = x^17 + ...
%e => a(4)=1, b(4)=17, GF - ... - x^17/(1-x)(1-x^2)(1-x^3)(1-x^4) = -x^21+...
%e => a(5)=-1, b(5)=21, GF - ... + x^21/... etc.
%o (PARI) p=1/prod(k=1,25,1-x^prime(k),1+O(x^999))/* Note: p1+...+p25 > 1000 */; for(k=0,99, print1(polcoeff(p,c=valuation(p,x)),",");p-=polcoeff(p,c)*x^c/prod(j=1,k,1-x^j,O(x^199)+1))
%Y Cf. A000607, A046676, A192541, A238804.
%K sign
%O 0,16
%A _M. F. Hasler_, Mar 06 2014
%E Example section corrected by _Vaclav Kotesovec_, Sep 12 2019
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