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%I #10 Apr 30 2013 11:18:02
%S 1,-4,10,-24,59,-146,360,-886,2182,-5376,13244,-32624,80364,-197968,
%T 487672,-1201319,2959297,-7289859,17957662,-44236464,108971015,
%U -268436517,661259918,-1628931424,4012669610,-9884711639,24349755585,-59982589144,147759635098
%N Convolutory inverse of the nonprimes.
%C Coefficients in 1/(1+g(x)), where g is the generating functions of the sequence of nonprimes: (1,4,6,8,9,...). For the convolutory inverse of the primes, see A030018. Conjecture: a(n+1)/a(n) has a limit, -2.4633754095588889..., analogous to the Backhouse constant.
%C The sequences with nonzero first term form a group under convolution. The identity is (1,0,0,0,...), and the inverse of a sequence r(1), r(2), r(3), ... is s(1), s(2), s(3),... given by s(1) = 1/r(1) and s(n) = -(r(2)*s(n-1) + ... + r(n)*s(1))/r(1). Thus, s(i) are the coefficients of the power series for 1/(r(1) + r(2)*x + r(3)*x^2 + ... ).
%H Clark Kimberling, <a href="/A225127/b225127.txt">Table of n, a(n) for n = 1..1000</a>
%e (1,4,6,8,9,...)**(1,-4,10,-24,59,...) = (1,0,0,0,0,...), where ** denotes convolution.
%t z = 1000; c = Complement[Range[z], Prime[Range[PrimePi[z]]]]; r[n_] := r[n] = c[[n]]; k[n_] := k[n] = 0; k[1] = 1; a[n_] := a[n] = (k[n] - Sum[r[i]*a[n - i + 1], {i, 2, n}])/r[1]; t = Table[a[n], {n, 1, 40}] (* A225127 *)
%Y Cf. A030018, A077607.
%K sign,easy
%O 1,2
%A _Clark Kimberling_, Apr 29 2013
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