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A294285
Sum of the larger parts of the partitions of n into two distinct parts with larger part squarefree.
2
0, 0, 2, 3, 3, 5, 11, 18, 18, 13, 23, 28, 28, 34, 48, 63, 63, 80, 80, 89, 89, 99, 121, 144, 144, 131, 157, 143, 143, 157, 187, 218, 218, 234, 268, 303, 303, 321, 359, 398, 398, 418, 460, 481, 481, 458, 504, 551, 551, 551, 551, 576, 576, 629, 629, 684, 684
OFFSET
1,3
COMMENTS
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
FORMULA
a(n) = Sum_{i=1..floor((n-1)/2)} (n - i) * mu(n - i)^2, where mu is the Möbius function (A008683).
a(n) = A211539(n + 1) - A294246(n). - David A. Corneth, Oct 27 2017
EXAMPLE
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
MATHEMATICA
Table[Sum[(n - i)*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 60}]
PROG
(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
(PARI) a(n) = sum(i=1, (n-1)\2, (n-i)*moebius(n-i)^2); \\ Michel Marcus, Nov 08 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wesley Ivan Hurt, Oct 26 2017
STATUS
approved