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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

One-half 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(n-i)^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(n-1) && n>1); forfactored(k=(n+1)\2, n-2, if(vecmax(k[2][, 2])==1 && issquarefree(n-k[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

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Last modified October 17 21:32 EDT 2019. Contains 328133 sequences. (Running on oeis4.)