|
%I #8 Oct 17 2012 19:38:50
%S 1,1,3,10,39,161,699,3135,14425,67694,322777,1559285,7615406,37539265,
%T 186525154,933239667,4697671339,23773865938,120889679621,617355432767,
%U 3164858856181,16281289560860,84023792421928,434886620261755,2256867537647996,11740881181554030
%N G.f. satisfies: A(x) = x + sum_{n>=1} A(x)^prime(n).
%F G.f.: A(x) = Series_Reversion(x - sum_{n>=1} x^prime(n)).
%F Let P(x) = Sum_{n>=1} x^prime(n) be the characteristic function of the primes, then the g.f. A(x) satisfies:
%F (1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) P(x)^n/n!,
%F (2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (P(x)^n/x)/n! ).
%e G.f.: A(x) = x + x^2 + 3*x^3 + 10*x^4 + 39*x^5 + 161*x^6 + 699*x^7 + 3135*x^8 +...
%e where
%e A(x) = x + A(x)^2 + A(x)^3 + A(x)^5 + A(x)^7 + A(x)^11 + A(x)^13 + A(x)^17 + A(x)^19 + A(x)^23 + A(x)^29 +...+ A(x)^prime(n) +...
%e Let P(x) = x^2 + x^3 + x^5 + x^7 + x^11 + x^13 +...+ x^prime(n) +... then
%e (1) A(x) = x + P(x) + d/dx P(x)^2/2! + d^2/dx^2 P(x)^3/3! + d^3/dx^3 P(x)^4/4! +...
%e (2) log(A(x)/x) = P(x)/x + d/dx (P(x)^2/x)/2! + d^2/dx^2 (P(x)^3/x)/3! + d^3/dx^3 (P(x)^4/x)/4! +...
%o (PARI) {a(n)=polcoeff(serreverse(x-sum(k=1,n,x^prime(k))+x*O(x^n)),n)}
%o for(n=1,25,print1(a(n),", "))
%o (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G));G}
%o {a(n)=local(A=x,P=sum(m=1,n,x^prime(m))+x*O(x^n)); A=x+sum(m=1, n, Dx(m-1, P^m/m!)+x*O(x^n)); polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%o (PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
%o {a(n)=local(A=x,P=sum(m=1,n,x^prime(m))+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, P^m/x/m!)+x*O(x^n))); polcoeff(A, n)}
%o for(n=1, 25, print1(a(n), ", "))
%K nonn
%O 1,3
%A _Paul D. Hanna_, Oct 17 2012
|
