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A309478
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Sum of the squarefree parts of the partitions of n into 3 parts.
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8
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0, 0, 0, 3, 4, 10, 14, 24, 32, 51, 56, 77, 95, 128, 146, 198, 225, 280, 318, 384, 416, 524, 549, 672, 726, 861, 915, 1063, 1143, 1292, 1382, 1551, 1661, 1867, 1966, 2211, 2355, 2618, 2762, 3094, 3263, 3602, 3798, 4185, 4409, 4869, 5078, 5524, 5794, 6264
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OFFSET
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0,4
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LINKS
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FORMULA
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a(n) = Sum_{j=1..floor(n/3)} Sum_{i=j..floor((n-j)/2)} (i * mu(i)^2 + j * mu(j)^2 + (n-i-j) * mu(n-i-j)^2), where mu is the Möbius function (A008683).
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EXAMPLE
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Figure 1: The partitions of n into 3 parts for n = 3, 4, ...
1+1+8
1+1+7 1+2+7
1+2+6 1+3+6
1+1+6 1+3+5 1+4+5
1+1+5 1+2+5 1+4+4 2+2+6
1+1+4 1+2+4 1+3+4 2+2+5 2+3+5
1+1+3 1+2+3 1+3+3 2+2+4 2+3+4 2+4+4
1+1+1 1+1+2 1+2+2 2+2+2 2+2+3 2+3+3 3+3+3 3+3+4 ...
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n | 3 4 5 6 7 8 9 10 ...
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a(n) | 3 4 10 14 24 32 51 56 ...
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MATHEMATICA
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Table[Sum[Sum[i*MoebiusMu[i]^2 + j*MoebiusMu[j]^2 + (n - i - j) MoebiusMu[n - i - j]^2, {i, j, Floor[(n - j)/2]}], {j, Floor[n/3]}], {n, 0, 50}]
Table[Total[Select[Flatten[IntegerPartitions[n, {3}]], SquareFreeQ]], {n, 0, 50}] (* Harvey P. Dale, Apr 09 2022 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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