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%I #7 Aug 27 2013 02:06:30
%S 1,2,3,4,5,4,7,8,3,9,11,4,13,11,14,8,17,4,19,4,1,4,23,24,5,17,27,22,
%T 29,16,31,32,24,4,30,36,37,4,36,10,41,18,43,41,17,27,47,40,28,29,7,10,
%U 53,9,1,24,49,4,59,57,61,35,31,48,39,16,67,24,51,9,71,46,73,4,56,11,55,62,79
%N Least positive integers that can appear as the coefficient of x^n the n-th iteration of an integer function F(x) where F(0)=0, for n>=1; the F(x) that satisfies this minimal condition is the g.f. of A119815.
%C For prime p, a(p) = p; for all n>=1, 0 < a(n) <=n.
%H Paul D. Hanna, <a href="/A119816/b119816.txt">Table of n, a(n) for n = 1..185</a>
%F a(n) = [x^n] F_n(x) where F_n(x) = F_{n-1}(F(x)) such that F(x) = g.f. of A119815 causes {a(n)} to be the least positive integers.
%e Let F(x) = g.f. of A119815 = [1,1,-1,1,1,-11,23,-20,731,-4860,...],
%e then the coefficient of x^n in the n-th iteration of F(x)
%e forms [1,2,3,4,5,4,7,8,3,9,11,...], as illustrated by:
%e F(x) = (1)x + x^2 - x^3 + x^4 + x^5 - 11x^6 + 23x^7 - 20x^8 + 731x^9+..
%e F(F(x)) = x + (2)x^2 - 2x^4 + 6x^5 - 8x^6 - 50x^7 + 78x^8 + 1688x^9+...
%e F(F(F(x))) = x + 3x^2 + (3)x^3 - 3x^4 - x^5 + 17x^6 - 81x^7 -370x^8+...
%e F(F(F(F(x)))) = x + 4x^2 + 8x^3 + (4)x^4 - 12x^5 + 4x^6 + 12x^7 +...
%e F(F(F(F(F(x))))) = x + 5x^2 + 15x^3 + 25x^4 + (5)x^5 - 55x^6 -33x^7+...
%e F(F(F(F(F(F(x)))))) = x + 6x^2 + 24x^3 + 66x^4 + 106x^5 + (4)x^6 +...
%o (PARI) {a(n)=local(A=vector(n),B,F=x+x^2,G);if(n==1|n==2,n,A[1]=1;A[2]=1;B=A;B[2]=2; for(m=3,n,G=x+x*O(x^n);for(k=1,m,G=subst(F,x,G)); B[m]=polcoeff(G,m,x);A[m]=(m-B[m])\m;F=F+A[m]*x^m);return(B[n]+n*A[n]))}
%Y Cf. A119815, A119818.
%K nonn
%O 1,2
%A _Paul D. Hanna_, May 31 2006
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