A294285 Sum of the larger parts of the partitions of n into two distinct parts with larger part squarefree.

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

Author

Wesley Ivan Hurt, Oct 26 2017

Keywords

• nonn
• easy
• changed

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

Examples

			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
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• Index entries for sequences related to partitions

Crossrefs

Cf. A005117, A008683, A066779, A211539, A262870.

Programs

Mathematica

Table[Sum[(n - i)*MoebiusMu[n - i]^2, {i, Floor[(n-1)/2]}], {n, 60}]
Table[Total[Select[IntegerPartitions[n,{2}],DuplicateFreeQ[#]&&SquareFreeQ[#[[1]]]&][[;;,1]]],{n,60}] (* Harvey P. Dale, Aug 30 2025 *)

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

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