A262992 Sum of the squarefree numbers among the partition parts of n into two parts.

0, 2, 3, 8, 6, 14, 17, 24, 24, 29, 34, 51, 45, 65, 72, 87, 87, 104, 104, 133, 123, 155, 166, 189, 189, 202, 215, 229, 215, 259, 274, 305, 305, 355, 372, 407, 407, 463, 482, 521, 521, 583, 604, 669, 647, 670, 693, 740, 740, 740, 740, 817, 791, 844, 844, 899

Offset

1,2

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Index entries for sequences related to partitions

Formula

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).

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

Example

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.
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.

Maple

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);

Mathematica

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

Prog

(PARI) vector(100, n, sum(k=1, n\2, k*moebius(k)^2 + (n-k)*moebius(n-k)^2)) \\ Altug Alkan, Oct 07 2015
(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

Crossrefs

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

Sequence in context: A112977 A120390 A109230 * A036970 A110144 A327353

Adjacent sequences: A262989 A262990 A262991 * A262993 A262994 A262995

Keyword

nonn,easy

Author

Wesley Ivan Hurt, Oct 06 2015

Status

approved