

A225127


Convolutory inverse of the nonprimes.


5



1, 4, 10, 24, 59, 146, 360, 886, 2182, 5376, 13244, 32624, 80364, 197968, 487672, 1201319, 2959297, 7289859, 17957662, 44236464, 108971015, 268436517, 661259918, 1628931424, 4012669610, 9884711639, 24349755585, 59982589144, 147759635098
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OFFSET

1,2


COMMENTS

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.
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(n1) + ... + 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 + ... ).


LINKS

Clark Kimberling, Table of n, a(n) for n = 1..1000


EXAMPLE

(1,4,6,8,9,...)**(1,4,10,24,59,...) = (1,0,0,0,0,...), where ** denotes convolution.


MATHEMATICA

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 *)


CROSSREFS

Cf. A030018, A077607.
Sequence in context: A079844 A080617 A080628 * A230954 A190169 A212330
Adjacent sequences: A225124 A225125 A225126 * A225128 A225129 A225130


KEYWORD

sign,easy


AUTHOR

Clark Kimberling, Apr 29 2013


STATUS

approved



