A262992 - OEIS

%I #26 Jan 13 2018 21:15:44

%S 0,2,3,8,6,14,17,24,24,29,34,51,45,65,72,87,87,104,104,133,123,155,

%T 166,189,189,202,215,229,215,259,274,305,305,355,372,407,407,463,482,

%U 521,521,583,604,669,647,670,693,740,740,740,740,817,791,844,844,899

%N Sum of the squarefree numbers among the partition parts of n into two parts.

%H Charles R Greathouse IV, <a href="/A262992/b262992.txt">Table of n, a(n) for n = 1..10000</a>

%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 + (n-i)*mu(n-i)^2, where mu is the Möebius function (A008683).

%F a(n) = A262870(n) + A262871(n).

%e a(3)=3; there is one partition of 3 into two parts: (2,1). The sum of the squarefree parts of this partition is 2+1=3, so a(3)=3.

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

%p with(numtheory): A262992:=n->add(i*mobius(i)^2 + (n-i)*mobius(n-i)^2, i=1..floor(n/2)): seq(A262992(n), n=1..100);

%t Table[Sum[i*MoebiusMu[i]^2 + (n - i)*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]

%o (PARI) vector(100, n, sum(k=1, n\2, k*moebius(k)^2 + (n-k)*moebius(n-k)^2)) \\ _Altug Alkan_, Oct 07 2015

%o (PARI) a(n)=my(s, k2, m=n-1); forsquarefree(k=1, sqrtint(m), k2=k[1]^2; s+= k2*binomial(m\k2+1, 2)*moebius(k)); s + (n%4==2 && issquarefree(n/2))*n/2 \\ _Charles R Greathouse IV_, Jan 13 2018

%Y Cf. A066779, A008683, A071068, A261985, A262351, A262868, A262870, A262871, A262991.

%K nonn,easy

%O 1,2

%A _Wesley Ivan Hurt_, Oct 06 2015