|
%I #41 Nov 29 2022 09:42:11
%S 0,1,1,2,2,3,3,3,3,4,4,5,5,6,6,6,6,6,6,7,7,8,8,8,8,9,9,10,10,11,11,11,
%T 11,12,12,12,12,13,13,13,13,14,14,15,15,16,16,16,16,16,16,17,17,17,17,
%U 17,17,18,18,19,19,20,20,20,20,21,21,22,22,23,23
%N Number of squarefree numbers appearing among the smaller parts of the partitions of n into two parts.
%C Number of distinct rectangles with integer length and squarefree width such that L + W = n, W <= L. For example, a(14) = 6; the rectangles are 13 X 1, 12 X 2, 11 X 3, 9 X 5, 8 X 6, 7 X 7. - _Wesley Ivan Hurt_, Nov 04 2017
%H Charles R Greathouse IV, <a href="/A262869/b262869.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor(n/2)} mu(i)^2, where mu is the Möebius function (A008683).
%F a(n) = A262991(n) - A262868(n).
%F a(n) = A013928(floor(n/2)+1). - _Georg Fischer_, Nov 29 2022
%e a(5)=2; there are two partitions of 5 into two parts: (4,1) and (3,2). Both of the smaller parts are squarefree, thus a(5)=2.
%e a(6)=3; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). Among the three smaller parts, all are squarefree, thus a(6)=3.
%p with(numtheory): A262869:=n->add(mobius(i)^2, i=1..floor(n/2)): seq(A262869(n), n=1..100);
%t Table[Sum[MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 100}]
%t Table[Count[IntegerPartitions[n,{2}][[All,2]],_?SquareFreeQ],{n,80}] (* _Harvey P. Dale_, Oct 17 2021 *)
%o (PARI) a(n) = sum(i=1, n\2, moebius(i)^2); \\ _Michel Marcus_, Oct 04 2015
%o (PARI) a(n)=my(s); n\=2; forsquarefree(k=1, sqrtint(n), s += n\k[1]^2*moebius(k)); s \\ _Charles R Greathouse IV_, Jan 08 2018
%Y Cf. A008683, A013928, A071068, A261985, A262351, A262868, A262870, A262871, A262991, A262992.
%K nonn,easy
%O 1,4
%A _Wesley Ivan Hurt_, Oct 03 2015
|
