A225127 Convolutory inverse of the nonprimes.

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

Offset: 1

Author

Clark Kimberling, Apr 29 2013

Keywords

• sign
• easy

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(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 + ... ).

Examples

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

Links

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

Crossrefs

Cf. A030018, A077607.

Programs

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 *)