A262991 - OEIS

A262991

Number of squarefree numbers among the parts of the partitions of n into two parts.

0, 2, 2, 4, 3, 5, 5, 6, 6, 7, 7, 9, 8, 10, 10, 11, 11, 12, 12, 14, 13, 15, 15, 16, 16, 17, 17, 18, 17, 19, 19, 20, 20, 22, 22, 23, 23, 25, 25, 26, 26, 28, 28, 30, 29, 30, 30, 31, 31, 31, 31, 33, 32, 33, 33, 34, 34, 36, 36, 38, 37, 39, 39, 39, 39, 41, 41, 43

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OFFSET

1,2

LINKS

Table of n, a(n) for n=1..68.

Index entries for sequences related to partitions

FORMULA

a(n) = Sum_{i=1..floor(n/2)} mu(i)^2 + mu(n-i)^2, where mu is the Möebius function (A008683).

a(n) = A262868(n) + A262869(n).

EXAMPLE

a(5)=3; there are 2 partitions of 5 into two parts: (4,1) and (3,2). Three of the parts in the partitions are squarefree, so a(5)=3.

a(6)=5; there are 3 partitions of 6 into two parts: (5,1), (4,2) and (3,3). Five of the parts in the partitions are squarefree, so a(6)=5.

MAPLE

with(numtheory): A262991:=n->add(mobius(i)^2+mobius(n-i)^2, i=1..floor(n/2)): seq(A262991(n), n=1..100);

MATHEMATICA

Table[Sum[MoebiusMu[i]^2 + MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]

Table[Count[Flatten[IntegerPartitions[n, {2}]], _?SquareFreeQ], {n, 70}] (* Harvey P. Dale, Aug 18 2021 *)

PROG

(PARI) vector(100, n, sum(k=1, n\2, moebius(k)^2 + moebius(n-k)^2)) \\ Altug Alkan, Oct 07 2015

CROSSREFS

Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262992.

Sequence in context: A372825 A225381 A007728 * A077026 A060026 A329437

Adjacent sequences: A262988 A262989 A262990 * A262992 A262993 A262994

KEYWORD

nonn,easy

AUTHOR

Wesley Ivan Hurt, Oct 06 2015

STATUS

approved