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A238804
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Powers in a variant of Ramanujan's wrong identity for prime number partitions.
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3
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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
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OFFSET
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0,2
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COMMENTS
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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, ...
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LINKS
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EXAMPLE
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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.
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PROG
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(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])
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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