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A262351
Sum of the parts in the partitions of n into exactly two squarefree parts.
11
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
1,2
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
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).
EXAMPLE
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.
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Sep 18 2015
STATUS
approved