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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. 4

%I #10 May 25 2017 13:15:53

%S 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,

%T 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,

%U 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

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

%H Antti Karttunen, <a href="/A076413/b076413.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%t 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)

%o (PARI)

%o 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_

%o A076413(n) = if(1==n,0,(A076078(n)/2 - 2^(numdiv(n)-2))); \\ _Antti Karttunen_, May 25 2017

%Y Cf. A000005, A076078.

%K nonn

%O 1,12

%A _Amarnath Murthy_ and _Dean Hickerson_, Oct 09 2002

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)