A238882 Coefficients in a variant of Ramanujan's wrong identity for prime number partitions.

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, -2, -8, -4, -5, -4, -1, 1, 2, 5, 6, 13, 12, 16, 18, 21, 25, 23, 30, 22, 23, 21, 21, 18, 14, 8, -1, -9, -20, -36, -36, -51, -61, -75, -80, -96, -103

Offset: 0

Author

M. F. Hasler, Mar 06 2014

Keywords

• sign

Comments

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).

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, ...

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+... (cf. A000607)
=> a(0)=1, b(0)=0, GF - 1 = x^2 + ....
=> a(1)=1, b(1)=2, GF - 1 - x^2/(1-x) = x^5 + ...
=> a(2)=1, b(2)=5, GF - 1 - x^2/(1-x) - x^5/(1-x)(1-x^2) = x^10 + ...
=> a(3)=1, b(3)=10, GF - ... - x^10/(1-x)(1-x^2)(1-x^3) = x^17 + ...
=> a(4)=1, b(4)=17, GF - ... - x^17/(1-x)(1-x^2)(1-x^3)(1-x^4) = -x^21+...
=> a(5)=-1, b(5)=21, GF - ... + x^21/... etc.
		

Crossrefs

Cf. A000607, A046676, A192541, A238804.

Programs

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))

Extensions

Example section corrected by Vaclav Kotesovec, Sep 12 2019