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A308844
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Sum of the second largest parts in the partitions of n into 5 squarefree parts.
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4
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0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 16, 18, 29, 33, 52, 59, 83, 93, 125, 138, 178, 196, 252, 275, 350, 380, 471, 506, 634, 689, 839, 901, 1096, 1176, 1405, 1484, 1767, 1861, 2199, 2294, 2695, 2823, 3281, 3388, 3941, 4101, 4714, 4901, 5607, 5843, 6643, 6893
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OFFSET
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0,8
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LINKS
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FORMULA
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a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l)^2 * i, where mu is the Möbius function (A008683).
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EXAMPLE
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The partitions of n into 5 parts for n = 10, 11, ..
1+1+1+1+10
1+1+1+2+9
1+1+1+3+8
1+1+1+4+7
1+1+1+5+6
1+1+1+1+9 1+1+2+2+8
1+1+1+2+8 1+1+2+3+7
1+1+1+3+7 1+1+2+4+6
1+1+1+4+6 1+1+2+5+5
1+1+1+5+5 1+1+3+3+6
1+1+1+1+8 1+1+2+2+7 1+1+3+4+5
1+1+1+2+7 1+1+2+3+6 1+1+4+4+4
1+1+1+3+6 1+1+2+4+5 1+2+2+2+7
1+1+1+1+7 1+1+1+4+5 1+1+3+3+5 1+2+2+3+6
1+1+1+2+6 1+1+2+2+6 1+1+3+4+4 1+2+2+4+5
1+1+1+3+5 1+1+2+3+5 1+2+2+2+6 1+2+3+3+5
1+1+1+1+6 1+1+1+4+4 1+1+2+4+4 1+2+2+3+5 1+2+3+4+4
1+1+1+2+5 1+1+2+2+5 1+1+3+3+4 1+2+2+4+4 1+3+3+3+4
1+1+1+3+4 1+1+2+3+4 1+2+2+2+5 1+2+3+3+4 2+2+2+2+6
1+1+2+2+4 1+1+3+3+3 1+2+2+3+4 1+3+3+3+3 2+2+2+3+5
1+1+2+3+3 1+2+2+2+4 1+2+3+3+3 2+2+2+2+5 2+2+2+4+4
1+2+2+2+3 1+2+2+3+3 2+2+2+2+4 2+2+2+3+4 2+2+3+3+4
2+2+2+2+2 2+2+2+2+3 2+2+2+3+3 2+2+3+3+3 2+3+3+3+3
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n | 10 11 12 13 14 ...
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a(n) | 10 16 18 29 33 ...
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MATHEMATICA
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Table[Total[Select[IntegerPartitions[n, {5}], AllTrue[#, SquareFreeQ]&][[All, 2]]], {n, 0, 60}] (* Harvey P. Dale, Nov 19 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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