Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #11 Apr 17 2021 13:04:55
%S 0,0,0,0,4,4,8,11,19,22,32,38,51,59,75,86,108,123,147,167,197,218,254,
%T 281,322,354,400,437,491,534,592,643,710,765,840,903,984,1055,1145,
%U 1222,1324,1410,1517,1614,1734,1837,1968,2083,2222,2348,2499,2633,2797
%N Number of squarefree parts in the partitions of n into 4 parts.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F 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).
%e Figure 1: The partitions of n into 4 parts for n = 8, 9, ..
%e 1+1+1+9
%e 1+1+2+8
%e 1+1+3+7
%e 1+1+4+6
%e 1+1+1+8 1+1+5+5
%e 1+1+2+7 1+2+2+7
%e 1+1+1+7 1+1+3+6 1+2+3+6
%e 1+1+2+6 1+1+4+5 1+2+4+5
%e 1+1+3+5 1+2+2+6 1+3+3+5
%e 1+1+1+6 1+1+4+4 1+2+3+5 1+3+4+4
%e 1+1+1+5 1+1+2+5 1+2+2+5 1+2+4+4 2+2+2+6
%e 1+1+2+4 1+1+3+4 1+2+3+4 1+3+3+4 2+2+3+5
%e 1+1+3+3 1+2+2+4 1+3+3+3 2+2+2+5 2+2+4+4
%e 1+2+2+3 1+2+3+3 2+2+2+4 2+2+3+4 2+3+3+4
%e 2+2+2+2 2+2+2+3 2+2+3+3 2+3+3+3 3+3+3+3
%e --------------------------------------------------------------------------
%e n | 8 9 10 11 12 ...
%e --------------------------------------------------------------------------
%e a(n) | 19 22 32 38 51 ...
%e --------------------------------------------------------------------------
%e - _Wesley Ivan Hurt_, Sep 07 2019
%t 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}]
%t Table[Count[Flatten[IntegerPartitions[n,{4}]],_?SquareFreeQ],{n,0,60}] (* _Harvey P. Dale_, Apr 17 2021 *)
%Y Cf. A008683.
%K nonn
%O 0,5
%A _Wesley Ivan Hurt_, Aug 03 2019