A262870 Sum of the squarefree numbers appearing among the larger parts of the partitions of n into two parts.

0, 1, 2, 5, 3, 8, 11, 18, 18, 18, 23, 34, 28, 41, 48, 63, 63, 80, 80, 99, 89, 110, 121, 144, 144, 144, 157, 157, 143, 172, 187, 218, 218, 251, 268, 303, 303, 340, 359, 398, 398, 439, 460, 503, 481, 481, 504, 551, 551, 551, 551, 602, 576, 629, 629, 684, 684

Offset: 1

Author

Wesley Ivan Hurt, Oct 03 2015

Keywords

• nonn
• easy

Examples

			a(4)=5; there are two partitions of 4 into two parts: (3,1) and (2,2). The sum of the larger squarefree parts is 3+2=5, thus a(4)=5.
a(5)=3; there are two partitions of 5 into two parts: (4,1) and (3,2). Of the larger parts, 3 is the only squarefree part, so a(5)=3.
		

Links

• Index entries for sequences related to partitions

Crossrefs

Cf. A008683, A071068, A261985, A262351, A262868, A262869, A262871, A262991, A262992.

Programs

Maple

with(numtheory): A262870:=n->add((n-i)*mobius(n-i)^2, i=1..floor(n/2)): seq(A262870(n), n=1..100);

Mathematica

Table[Sum[(n - i) MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 70}]

PARI

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

PARI

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

Formula

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

a(n) = A262992(n) - A262871(n).