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
LINKS
FORMULA
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