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%I #23 Sep 18 2021 14:13:38
%S 0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,
%T 5,5,6,6,6,6,7,7,7,7,7,7,7,7,8,8,9,9,9,9,10,10,11,11,11,11,11,11,11,
%U 11,12,12,12,12,12,12,12,12,13,13,13,13,13,13,13
%N Number of partitions of n into two parts such that the smaller part is nonsquarefree.
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor(n/2)} (1 - mu(i)^2), where mu is the Möbius function (A008683).
%F a(2*n) = a(2*n + 1) for n >= 0. - _David A. Corneth_, Oct 25 2017
%e The nonsquarefree numbers up to 10 are 4, 8 and 9. So a(n) = 0 for n = 0 to 2*4 - 1 = 7, a(n) = 1 for n = 2*4 to 2*8 - 1, a(n) = 2 for n = 2*8 = 16 to 2*9 - 1 = 17. We haven't filled anything in yet for n = 18 to 2 * 10 = so a(n) = 3 for n = 18 to 20. We haven't checked for nonsquarefree numbers up for n > 10 so stop here. - _David A. Corneth_, Oct 25 2017
%t Table[Sum[(1 - MoebiusMu[k]^2), {k, Floor[n/2]}], {n, 200}]
%t Table[Count[IntegerPartitions[n,{2}],_?(!SquareFreeQ[#[[2]]]&)],{n,0,80}] (* _Harvey P. Dale_, Sep 18 2021 *)
%o (PARI) first(n) = {my(res = vector(n), nsqrfr = List(), t = 0); for(i = 2, sqrtint(n\2), for(k = 1, (n\2) \ i^2, listput(nsqrfr, k * i^2))); listsort(nsqrfr, 1); for(i = 1, #nsqrfr, for(j = t, nsqrfr[i] - 1, for(k = 1, 2, res[2*j + k] = i-1)); t = nsqrfr[i]); for(i=2*t+1, n, res[i] = res[2*t] + 1); res} \\ _David A. Corneth_, Oct 25 2017
%Y Cf. A008683, A008966, A013929, A294235.
%K nonn,easy
%O 0,17
%A _Wesley Ivan Hurt_, Oct 25 2017
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