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%I #33 Jun 20 2019 04:25:40
%S 0,0,2,3,3,5,11,18,18,13,23,28,28,34,48,63,63,80,80,89,89,99,121,144,
%T 144,131,157,143,143,157,187,218,218,234,268,303,303,321,359,398,398,
%U 418,460,481,481,458,504,551,551,551,551,576,576,629,629,684,684
%N Sum of the larger parts of the partitions of n into two distinct parts with larger part squarefree.
%C Sum of the lengths of the distinct rectangles with squarefree length and positive integer width such that L + W = n, W < L. For example, a(14) = 34; the rectangles are 1 X 13, 3 X 11, 4 X 10. The sum of the lengths is then 13 + 11 + 10 = 34. - _Wesley Ivan Hurt_, Nov 01 2017
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%F a(n) = Sum_{i=1..floor((n-1)/2)} (n - i) * mu(n - i)^2, where mu is the Möbius function (A008683).
%F a(n) = A211539(n + 1) - A294246(n). - _David A. Corneth_, Oct 27 2017
%e 10 can be partitioned into two distinct parts as follows: (1, 9), (2, 8), (3, 7), (4, 6). The squarefree larger parts are 6 and 7, which sum to a(10) = 13. - _David A. Corneth_, Oct 27 2017
%t Table[Sum[(n - i)*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 60}]
%o (PARI) first(n) = {my(res = vector(n, i, binomial(i, 2) - binomial(i\2+1, 2)), nsqrfr = List()); forprime(i=2, sqrtint(n), for(k = 1, n \ i^2, listput(nsqrfr, k*i^2))); listsort(nsqrfr, 1); for(i=1, #nsqrfr, for(m = nsqrfr[i]+1, min(2*nsqrfr[i]-1, n), res[m]-=nsqrfr[i])); res} \\ _David A. Corneth_, Oct 27 2017
%o (PARI) a(n) = sum(i=1, (n-1)\2, (n-i)*moebius(n-i)^2); \\ _Michel Marcus_, Nov 08 2017
%Y Cf. A005117, A008683, A066779, A211539, A262870.
%K nonn,easy
%O 1,3
%A _Wesley Ivan Hurt_, Oct 26 2017
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