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A119816
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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.
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3
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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, 29, 16, 31, 32, 24, 4, 30, 36, 37, 4, 36, 10, 41, 18, 43, 41, 17, 27, 47, 40, 28, 29, 7, 10, 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
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OFFSET
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1,2
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COMMENTS
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For prime p, a(p) = p; for all n>=1, 0 < a(n) <=n.
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LINKS
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FORMULA
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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.
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EXAMPLE
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Let F(x) = g.f. of A119815 = [1,1,-1,1,1,-11,23,-20,731,-4860,...],
then the coefficient of x^n in the n-th iteration of F(x)
forms [1,2,3,4,5,4,7,8,3,9,11,...], as illustrated by:
F(x) = (1)x + x^2 - x^3 + x^4 + x^5 - 11x^6 + 23x^7 - 20x^8 + 731x^9+..
F(F(x)) = x + (2)x^2 - 2x^4 + 6x^5 - 8x^6 - 50x^7 + 78x^8 + 1688x^9+...
F(F(F(x))) = x + 3x^2 + (3)x^3 - 3x^4 - x^5 + 17x^6 - 81x^7 -370x^8+...
F(F(F(F(x)))) = x + 4x^2 + 8x^3 + (4)x^4 - 12x^5 + 4x^6 + 12x^7 +...
F(F(F(F(F(x))))) = x + 5x^2 + 15x^3 + 25x^4 + (5)x^5 - 55x^6 -33x^7+...
F(F(F(F(F(F(x)))))) = x + 6x^2 + 24x^3 + 66x^4 + 106x^5 + (4)x^6 +...
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PROG
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(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]))}
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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