A262351 Sum of the parts in the partitions of n into exactly two squarefree parts.

0, 2, 3, 8, 5, 12, 14, 24, 18, 20, 22, 48, 39, 42, 45, 80, 68, 72, 57, 120, 84, 110, 92, 168, 125, 130, 135, 196, 145, 150, 155, 256, 198, 238, 210, 396, 259, 266, 273, 440, 328, 336, 387, 572, 450, 368, 376, 624, 490, 400, 357, 728, 530, 540, 385, 728, 570

Offset: 1

Author

Wesley Ivan Hurt, Sep 18 2015

Keywords

• nonn
• easy

Comments

One-half of the sum of the perimeters of the rectangles with squarefree length and width such that L + W = n, W <= L. For example, a(8) = 24; the rectangles are 1 X 7, 2 X 6 and 3 X 5 with perimeters 16, 16 and 16. Then (16 + 16 + 16)/2 = 48/2 = 24. - Wesley Ivan Hurt, Nov 04 2017

Examples

			a(4) = 8; There are two partitions of 4 into two squarefree parts: (3,1) and (2,2). Thus we have a(4) = (3+1) + (2+2) = 8.
a(7) = 14; There are three partitions of 7 into two parts: (6,1), (5,2) and (4,3). Since only two of these partitions have squarefree parts, we have a(7) = (6+1) + (5+2) = 14.
		

Links

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

Crossrefs

Cf. A005117, A008683, A071068.

Programs

Mathematica

Table[n*Sum[MoebiusMu[i]^2*MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 70}]
Table[Total[Flatten[Select[IntegerPartitions[n,{2}],AllTrue[ #,SquareFreeQ]&]]],{n,60}] (* Harvey P. Dale, Aug 19 2021 *)

PARI

a(n)=my(s=issquarefree(n-1) && n>1); forfactored(k=(n+1)\2,n-2, if(vecmax(k[2][,2])==1 && issquarefree(n-k[1]), s++)); s*n \\ Charles R Greathouse IV, Nov 05 2017

Formula

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

a(n) = n * A071068(n).