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Sum of the squarefree numbers appearing among the smaller parts of the partitions of n into two parts.
7

%I #34 Feb 10 2018 21:55:54

%S 0,1,1,3,3,6,6,6,6,11,11,17,17,24,24,24,24,24,24,34,34,45,45,45,45,58,

%T 58,72,72,87,87,87,87,104,104,104,104,123,123,123,123,144,144,166,166,

%U 189,189,189,189,189,189,215,215,215,215,215,215,244,244,274

%N Sum of the squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>

%F a(n) = Sum_{i=1..floor(n/2)} i * mu(i)^2, where mu is the Möebius function (A008683).

%F a(n) = A262992(n) - A262870(n).

%e a(5)=3; there are two partitions of 5 into two parts: (4,1) and (3,2). The sum of the smaller squarefree parts is 1+2=3. Thus a(5)=3.

%e a(6)=6; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). All of the smaller parts are squarefree, so a(6) = 1+2+3 = 6.

%p with(numtheory): A262871:=n->add(i*mobius(i)^2, i=1..floor(n/2)): seq(A262871(n), n=1..100);

%t Table[Sum[i*MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 70}]

%o (PARI) a(n) = sum(i=1, n\2, i * moebius(i)^2); \\ _Michel Marcus_, Oct 04 2015

%o (PARI) a(n)=my(s); forsquarefree(k=1,n\2, s += k[1]); s \\ _Charles R Greathouse IV_, Jan 08 2018

%Y Cf. A008683, A071068, A261985, A262351, A262868, A262869, A262870, A262991, A262992.

%K nonn,easy

%O 1,4

%A _Wesley Ivan Hurt_, Oct 03 2015