A339638 - OEIS

%I #10 Dec 25 2020 10:37:34

%S 1,3,5,9,11,21,23,31,35,45,47,91,93,103,113,129,131,175,177,221,231,

%T 241,243,427,431,441,449,493,495,713,715,747,757,767,777,1177,1179,

%U 1189,1199,1383,1385,1603,1605,1649,1693,1703,1705,2457,2461,2505,2515,2559,2561,2745,2755

%N Number of nonempty sets of distinct positive integers that have a least common multiple <= n.

%C Partial sums of A076078.

%H Vaclav Kotesovec, <a href="/A339638/b339638.txt">Table of n, a(n) for n = 1..10000</a>

%H <a href="/index/Lc#lcm">Index entries for sequences related to lcm's</a>

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

%e a(5) = 11 sets: {1}, {2}, {3}, {4}, {5}, {1, 2}, {1, 3}, {1, 4}, {1, 5}, {2, 4} and {1, 2, 4}.

%t Table[Sum[Sum[MoebiusMu[k/d] (2^DivisorSigma[0, d] - 1), {d, Divisors[k]}], {k, n}], {n, 55}]

%t Accumulate[Table[Sum[MoebiusMu[k/d] (2^DivisorSigma[0, d] - 1), {d, Divisors[k]}], {k, 1, 60}]] (* _Vaclav Kotesovec_, Dec 25 2020 *)

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, moebius(k/d) * (2^numdiv(d) - 1))); \\ _Michel Marcus_, Dec 11 2020

%Y Cf. A000005, A076078, A100587.

%K nonn

%O 1,2

%A _Ilya Gutkovskiy_, Dec 11 2020