A339667 - OEIS

%I #8 Dec 15 2021 17:32:28

%S 0,1,1,3,1,5,1,7,3,5,1,19,1,5,5,15,1,19,1,19,5,5,1,71,3,5,7,19,1,37,1,

%T 31,5,5,5,111,1,5,5,71,1,37,1,19,19,5,1,271,3,19,5,19,1,71,5,71,5,5,1,

%U 347,1,5,19,63,5,37,1,19,5,37,1,703,1,5,19,19,5,37,1,271

%N Number of nonempty subsets of divisors of n having a common factor > 1.

%H Antti Karttunen, <a href="/A339667/b339667.txt">Table of n, a(n) for n = 1..20000</a>

%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>

%F a(n) = -Sum_{d|n, d < n} mu(n/d) * (2^tau(d) - 1), where tau = A000005, and mu = A008683.

%F a(n) = A100587(n) - A076078(n).

%F a(p) = 1 for p prime.

%e a(12) = 19 subsets: {2}, {3}, {4}, {6}, {12}, {2, 4}, {2, 6}, {2, 12}, {3, 6}, {3, 12}, {4, 6}, {4, 12}, {6, 12}, {2, 4, 6}, {2, 4, 12}, {2, 6, 12}, {3, 6, 12}, {4, 6, 12} and {2, 4, 6, 12}.

%t Table[-DivisorSum[n, MoebiusMu[n/#] (2^DivisorSigma[0, #] - 1) &, # < n &], {n, 80}]

%o (PARI) A339667(n) = -sumdiv(n, d, if(d==n,0, moebius(n/d)*((2^numdiv(d))-1))); \\ _Antti Karttunen_, Dec 15 2021

%Y Cf. A000005, A008683, A027750, A076078, A100587, A109511.

%K nonn

%O 1,4

%A _Ilya Gutkovskiy_, Dec 12 2020