

A262351


Sum of the parts in the partitions of n into exactly two squarefree parts.


11



0, 2, 3, 8, 5, 12, 14, 24, 18, 20, 22, 48, 39, 42, 45, 80, 68, 72, 57, 120, 84, 110, 92, 168, 125, 130, 135, 196, 145, 150, 155, 256, 198, 238, 210, 396, 259, 266, 273, 440, 328, 336, 387, 572, 450, 368, 376, 624, 490, 400, 357, 728, 530, 540, 385, 728, 570
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OFFSET

1,2


COMMENTS

Onehalf of the sum of the perimeters of the rectangles with squarefree length and width such that L + W = n, W <= L. For example, a(8) = 24; the rectangles are 1 X 7, 2 X 6 and 3 X 5 with perimeters 16, 16 and 16. Then (16 + 16 + 16)/2 = 48/2 = 24.  Wesley Ivan Hurt, Nov 04 2017


LINKS

Peter Kagey, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions


FORMULA

a(n) = n * Sum_{i=1..floor(n/2)} mu(i)^2 * mu(ni)^2, where mu is the Möbius function (A008683).
a(n) = n * A071068(n).


EXAMPLE

a(4) = 8; There are two partitions of 4 into two squarefree parts: (3,1) and (2,2). Thus we have a(4) = (3+1) + (2+2) = 8.
a(7) = 14; There are three partitions of 7 into two parts: (6,1), (5,2) and (4,3). Since only two of these partitions have squarefree parts, we have a(7) = (6+1) + (5+2) = 14.


MATHEMATICA

Table[n*Sum[MoebiusMu[i]^2*MoebiusMu[n  i]^2, {i, Floor[n/2]}], {n, 70}]


PROG

(PARI) a(n)=my(s=issquarefree(n1) && n>1); forfactored(k=(n+1)\2, n2, if(vecmax(k[2][, 2])==1 && issquarefree(nk[1]), s++)); s*n \\ Charles R Greathouse IV, Nov 05 2017


CROSSREFS

Cf. A005117, A008683, A071068.
Sequence in context: A066959 A086471 A249154 * A294211 A097505 A095168
Adjacent sequences: A262348 A262349 A262350 * A262352 A262353 A262354


KEYWORD

nonn,easy


AUTHOR

Wesley Ivan Hurt, Sep 18 2015


STATUS

approved



