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A309456
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Number of squarefree parts in the partitions of n into 4 parts.
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1
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0, 0, 0, 0, 4, 4, 8, 11, 19, 22, 32, 38, 51, 59, 75, 86, 108, 123, 147, 167, 197, 218, 254, 281, 322, 354, 400, 437, 491, 534, 592, 643, 710, 765, 840, 903, 984, 1055, 1145, 1222, 1324, 1410, 1517, 1614, 1734, 1837, 1968, 2083, 2222, 2348, 2499, 2633, 2797
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OFFSET
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0,5
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LINKS
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FORMULA
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a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} (mu(i)^2 + mu(j)^2 + mu(k)^2 + mu(n-i-j-k)^2), where mu is the Möbius function (A008683).
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EXAMPLE
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Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
1+1+1+9
1+1+2+8
1+1+3+7
1+1+4+6
1+1+1+8 1+1+5+5
1+1+2+7 1+2+2+7
1+1+1+7 1+1+3+6 1+2+3+6
1+1+2+6 1+1+4+5 1+2+4+5
1+1+3+5 1+2+2+6 1+3+3+5
1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4
1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6
1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5
1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4
1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4
2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3
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n | 8 9 10 11 12 ...
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a(n) | 19 22 32 38 51 ...
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MATHEMATICA
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Table[Sum[Sum[Sum[(MoebiusMu[i]^2 + MoebiusMu[j]^2 + MoebiusMu[k]^2 + MoebiusMu[n - i - j - k]^2), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
Table[Count[Flatten[IntegerPartitions[n, {4}]], _?SquareFreeQ], {n, 0, 60}] (* Harvey P. Dale, Apr 17 2021 *)
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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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