|
%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
|
