A262871 Sum of the squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

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, 58, 72, 72, 87, 87, 87, 87, 104, 104, 104, 104, 123, 123, 123, 123, 144, 144, 166, 166, 189, 189, 189, 189, 189, 189, 215, 215, 215, 215, 215, 215, 244, 244, 274

Offset: 1

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

• nonn
• easy

Examples

			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.
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.
		

Links

• Index entries for sequences related to partitions

Crossrefs

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

Programs

Maple

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

Mathematica

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

PARI

a(n) = sum(i=1, n\2, i * moebius(i)^2); \\ Michel Marcus, Oct 04 2015

PARI

a(n)=my(s); forsquarefree(k=1,n\2, s += k[1]); s \\ Charles R Greathouse IV, Jan 08 2018

Formula

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

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