A218001 G.f. satisfies: A(x) = x + sum_{n>=1} A(x)^prime(n).

1, 1, 3, 10, 39, 161, 699, 3135, 14425, 67694, 322777, 1559285, 7615406, 37539265, 186525154, 933239667, 4697671339, 23773865938, 120889679621, 617355432767, 3164858856181, 16281289560860, 84023792421928, 434886620261755, 2256867537647996, 11740881181554030

Offset

1,3

Links

Table of n, a(n) for n=1..26.

Formula

G.f.: A(x) = Series_Reversion(x - sum_{n>=1} x^prime(n)).

Let P(x) = Sum_{n>=1} x^prime(n) be the characteristic function of the primes, then the g.f. A(x) satisfies:

(1) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) P(x)^n/n!,

(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) (P(x)^n/x)/n! ).

Example

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 +...
where
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) +...
Let P(x) = x^2 + x^3 + x^5 + x^7 + x^11 + x^13 +...+ x^prime(n) +... then
(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! +...
(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! +...

Prog

(PARI) {a(n)=polcoeff(serreverse(x-sum(k=1, n, x^prime(k))+x*O(x^n)), n)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{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)}
for(n=1, 25, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(G=F); for(i=1, n, G=deriv(G)); G}
{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)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Sequence in context: A151071 A063022 A371713 * A307490 A253194 A367258

Adjacent sequences: A217998 A217999 A218000 * A218002 A218003 A218004

Keyword

nonn

Author

Paul D. Hanna, Oct 17 2012

Status

approved