A076413 Number of ways n is expressible as the least common multiple of a nonempty set of positive integers not containing either 1 or n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 6, 0, 1, 1, 0, 0, 6, 0, 6, 1, 1, 0, 28, 0, 1, 0, 6, 0, 45, 0, 0, 1, 1, 1, 72, 0, 1, 1, 28, 0, 45, 0, 6, 6, 1, 0, 120, 0, 6, 1, 6, 0, 28, 1, 28, 1, 1, 0, 850, 0, 1, 6, 0, 1, 45, 0, 6, 1, 45, 0, 672, 0, 1, 6, 6, 1, 45, 0, 120, 0, 1, 0, 850, 1, 1, 1, 28, 0, 850, 1, 6, 1

Offset: 1

Author

Amarnath Murthy and Dean Hickerson, Oct 09 2002

Keywords

• nonn

Examples

			a(12)=6; the 6 sets are: {3,4}, {4,6}, {2,3,4}, {2,4,6}, {3,4,6}, {2,3,4,6}.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A076078.

Programs

Mathematica

a076078[n_] := Module[{d, i}, d=Divisors[n]; Sum[MoebiusMu[n/d[[i]]]*(2^DivisorSigma[0, d[[i]]]-1), {i, 1, Length[d]}]]; a[n_] := a076078[n]/2-2^(DivisorSigma[0, n]-2)

PARI

A076078(n) = { local(f, l, s, t, q); f = factor(n); l = matsize(f)[1]; s = 0; forvec(v = vector(l, i, [0, 1]), q = sum(i = 1, l, v[i]); t = (-1)^(l - q)*2^prod(i = 1, l, f[i, 2] + v[i]); s += t); s; } \\ This function from David Wasserman
A076413(n) = if(1==n,0,(A076078(n)/2 - 2^(numdiv(n)-2))); \\ Antti Karttunen, May 25 2017

Formula

a(n) = A076078(n)/2 - 2^(d(n)-2), where d(n)=A000005(n) is the number of divisors of n.