OFFSET
1,4
COMMENTS
Conjecture: the radius of convergence r satisfies A(r) = 1 where r = 0.35626109971368949500932502331626200111344438973344869586... Setting y = r and repeating y = y + y^prime(n) as n = 1,2,3,... yields y = 1 as the limit.
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1000
EXAMPLE
G.f.: A(x) = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 16*x^7 + 52*x^8 + 148*x^9 + 364*x^10 + 861*x^11 + 1983*x^12 + 4458*x^13 + 10260*x^14 + ...
A(x) is the limit of the composition of functions F_n(x) = (x + x^prime(n)), starting with F_1(x) = (x + x^2) and repeating F_{n+1}(x) = (x + x^prime(n+1)) o F_n(x) as illustrated below.
F_2(x) = (x + x^3) o (x + x^2) = x + x^2 + x^3 + 3*x^4 + 3*x^5 + x^6;;
F_3(x) = (x + x^5) o F_2(x) = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 15*x^7 + ...
F_4(x) = (x + x^7) o F_3(x) = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 16*x^7 + 52*x^8 + 148*x^9 + 364*x^10 + 860*x^11 + ...
F_5(x) = (x + x^11) o (x + x^7) o (x + x^5) o (x + x^3) o (x + x^2) = x + x^2 + x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 16*x^7 + 52*x^8 + 148*x^9 + 364*x^10 + 861*x^11 + 1983*x^12 + 4457*x^13 + ...
etc.
SPECIFIC VALUES.
A(t) = 3/4 at t = 0.35226701108274629620754794103015391763342134182729...
set y = t and repeat y = y + y^prime(n) as n = 1,2,3,... to obtain y = 3/4 as the limit.
A(t) = 1/2 at t = 0.30853993610294250373154036702973294716694137226555...
set y = t and repeat y = y + y^prime(n) as n = 1,2,3,... to obtain y = 1/2 as the limit.
A(1/3) = 0.600586726870158877232775281221044920811581450...
A(3/10) = 0.472847713007847815783627703497743920272388194...
A(1/4) = 0.348391858898138779991484272786295149803690300...
A(1/5) = 0.254947760061049750554066043265996958409227405...
A(1/6) = 0.202144563847561508378654945989810496437287923...
PROG
(PARI) {a(n) = my(F=x); for(k=1, n, F = subst(x + x^prime(k), x, F +x*O(x^n)); ); polcoef(GF=F, n)}
{upto(n) = a(n); Vec(GF)}
upto(40)
CROSSREFS
KEYWORD
nonn,new
AUTHOR
Paul D. Hanna, Jul 12 2026
STATUS
approved