A294101 -id:A294101 - cp's OEIS frontend

A071068 Number of ways to write n as a sum of two unordered squarefree numbers.

0, 1, 1, 2, 1, 2, 2, 3, 2, 2, 2, 4, 3, 3, 3, 5, 4, 4, 3, 6, 4, 5, 4, 7, 5, 5, 5, 7, 5, 5, 5, 8, 6, 7, 6, 11, 7, 7, 7, 11, 8, 8, 9, 13, 10, 8, 8, 13, 10, 8, 7, 14, 10, 10, 7, 13, 10, 11, 9, 15, 11, 11, 11, 15, 11, 11, 11, 18, 12, 13, 11, 21, 13, 14, 13, 20, 14, 13, 14, 20, 16, 13, 13, 22, 15

Offset: 1

Benoit Cloitre, May 26 2002

The natural density of the squarefree numbers is 6/Pi^2, so An < a(n) < Bn for all large enough n with A < 6/Pi^2 - 1/2 and B > 3/Pi^2. The Schnirelmann density of the squarefree numbers is 53/88 > 1/2, and so a(n) > 0 for all n > 1 (in fact, a(n+1) >= 9n/88). It follows from Theoreme 3 bis. in Cohen, Dress, & El Marraki along with finite checking up to 16089908 that 0.10792n < a(n) < 0.303967n for n > 36. (The lower bound holds for n > 1.) - Charles R Greathouse IV, Feb 02 2016

			12=1+11=2+10=5+7=6+6 hence a(12)=4.

T. D. Noe, Table of n, a(n) for n = 1..10000
Henri Cohen, Francois Dress, and Mahomed El Marraki, Explicit estimates for summatory functions linked to the Möbius μ-function, Funct. Approx. Comment. Math. 37:1 (2007), pp. 51-63.

Cf. A005117, A008683 (mu), A008966, A098235, A262991, A294101.

Table[Sum[Abs[MoebiusMu[i] MoebiusMu[n - i]],{i, 1, Floor[n/2]}],{n, 1, 85}] (* Indranil Ghosh, Mar 10 2017 *)
Table[Count[IntegerPartitions[n,{2}],?(AllTrue[#,SquareFreeQ]&)],{n,90}] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale, Aug 13 2020 *)

a(n)=sum(i=1,n\2,issquarefree(i)&&issquarefree(n-i)) \\ Charles R Greathouse IV, May 21 2013

list(lim)=my(n=lim\1); concat(0, ceil(Vec((Polrev(vector(n, k, issquarefree(k-1))) + O('x^(n+1)))^2)/2)) \\ Charles R Greathouse IV, May 21 2013

a(n)=my(s); forsquarefree(k=1,n\2, issquarefree(n-k[1]) && s++); s \\ Charles R Greathouse IV, Dec 20 2024

a(n) = Sum_{k=1..floor(n/2)} mu(k)^2 * mu(n-k)^2. - Wesley Ivan Hurt, May 20 2013

a(n) = (A262991(n) - A294101(n))/2. - Wesley Ivan Hurt, Jul 16 2025

A262991 Number of squarefree numbers among the parts of the partitions of n into two parts.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Oct 06 2015

			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.

Index entries for sequences related to partitions

Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262992, A294101.

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

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 *)

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

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).
a(n) = A294101(n) + 2*A071068(n). - Wesley Ivan Hurt, Jul 16 2025

A294100 Number of partitions of n into two squarefree parts or two nonsquarefree positive integer parts.

Original entry on oeis.org

0, 1, 1, 2, 1, 2, 2, 4, 2, 2, 2, 5, 4, 3, 3, 7, 5, 5, 3, 8, 5, 6, 4, 10, 6, 6, 6, 10, 7, 6, 6, 12, 8, 9, 7, 17, 9, 8, 8, 16, 10, 9, 11, 18, 13, 9, 9, 19, 13, 10, 8, 21, 14, 14, 8, 20, 14, 15, 11, 22, 15, 14, 14, 23, 15, 14, 14, 27, 16, 17, 13, 33, 17, 18, 16

Offset: 1

Wesley Ivan Hurt, Oct 22 2017

Number of distinct rectangles with both length and width squarefree or both nonsquarefree positive integers such that L + W = n, W <= L. For example, a(12) = 5; the rectangles are 1 X 11, 2 X 10, 4 X 8, 5 X 7, 6 X 6. - Wesley Ivan Hurt, Nov 04 2017

Index entries for sequences related to partitions

Cf. A008966, A294101.

Table[Sum[KroneckerDelta[MoebiusMu[k]^2, MoebiusMu[n - k]^2], {k, Floor[n/2]}], {n, 80}]

a(n) = sum(i=1, n\2, issquarefree(i) == issquarefree(n-i)); \\ Michel Marcus, Nov 08 2017

a(n) = Sum_{i=1..floor(n/2)} [A008966(i) = A008966(n-i)], where [] is the Iverson bracket.

A294232 Number of partitions of n into two parts with smaller part squarefree and larger part nonsquarefree.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 0, 1, 2, 2, 1, 2, 3, 3, 1, 2, 2, 3, 1, 3, 3, 4, 1, 3, 4, 4, 3, 5, 6, 6, 3, 5, 5, 6, 1, 5, 6, 6, 2, 5, 6, 5, 2, 5, 8, 8, 3, 6, 8, 9, 3, 7, 7, 10, 4, 7, 7, 9, 4, 8, 9, 9, 5, 9, 10, 10, 4, 10, 10, 12, 2, 10, 10, 11, 5, 11, 13, 12, 6

Offset: 1

Wesley Ivan Hurt, Oct 25 2017

Index entries for sequences related to partitions

Cf. A008683, A008966, A294101, A294233.

Table[Sum[(1 - MoebiusMu[n - k]^2) MoebiusMu[k]^2, {k, Floor[n/2]}], {n, 80}]

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

a(n) = A294101(n) - A294233(n). - Wesley Ivan Hurt, Jul 16 2025

A294233 Number of partitions of n into two parts with smaller part nonsquarefree and larger part squarefree.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 2, 3, 1, 2, 2, 3, 1, 3, 3, 3, 1, 2, 3, 3, 1, 3, 3, 4, 0, 4, 5, 5, 2, 5, 6, 5, 2, 4, 6, 6, 2, 5, 7, 8, 2, 5, 6, 9, 4, 7, 7, 9, 4, 7, 8, 8, 4, 8, 9, 9, 3, 8, 8, 10, 1, 9, 9, 10, 3, 8, 10, 9, 4, 8, 11, 11, 3, 9

Offset: 1

Wesley Ivan Hurt, Oct 25 2017

Index entries for sequences related to partitions

Cf. A008683, A008966, A294101, A294232.

Table[Sum[(1 - MoebiusMu[k]^2) MoebiusMu[n - k]^2, {k, Floor[n/2]}], {n, 80}]

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

a(n) = A294101(n) - A294232(n). - Wesley Ivan Hurt, Jul 16 2025

cp's OEIS Frontend

A071068 Number of ways to write n as a sum of two unordered squarefree numbers.

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A262991 Number of squarefree numbers among the parts of the partitions of n into two parts.

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A294100 Number of partitions of n into two squarefree parts or two nonsquarefree positive integer parts.

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A294232 Number of partitions of n into two parts with smaller part squarefree and larger part nonsquarefree.

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A294233 Number of partitions of n into two parts with smaller part nonsquarefree and larger part squarefree.

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