A294104 Sum of the differences of the larger and smaller parts in the partitions of n into two squarefree parts or two nonsquarefree parts.

0, 0, 1, 2, 1, 4, 8, 12, 8, 4, 10, 24, 22, 20, 29, 54, 33, 40, 31, 72, 37, 70, 54, 114, 68, 62, 74, 116, 97, 62, 114, 182, 136, 126, 147, 292, 175, 134, 180, 330, 214, 182, 227, 412, 273, 190, 209, 460, 343, 202, 156, 468, 392, 380, 156, 518, 394, 490, 303

Offset: 1

Author

Wesley Ivan Hurt, Oct 22 2017

Keywords

• nonn
• easy
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to partitions

Crossrefs

Cf. A008966, A294100.

Programs

Maple

f:= n -> add(n-2*i, i = select(t -> numtheory:-issqrfree(t)=numtheory:-issqrfree(n-t), [$1..n/2])):
map(f, [$1..100]); # Robert Israel, Jun 12 2020

Mathematica

Table[Sum[(n - 2 k) KroneckerDelta[MoebiusMu[k]^2, MoebiusMu[ n - k]^2], {k, Floor[n/2]}], {n, 80}]

Formula

a(n) = Sum_{i=1..floor(n/2)} (n - 2i) * [A008966(i) = A008966(n-i)], where [] is the Iverson bracket.