A262870 -id:A262870 - cp's OEIS frontend

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

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

A262868 Number of squarefree numbers appearing among the larger parts of the partitions of n into two parts.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Oct 03 2015

Number of distinct rectangles with squarefree length and integer width such that L + W = n, W <= L. For example, a(14) = 4; the rectangles are 1 X 13, 3 X 11, 4 X 10 and 7 X 7. - Wesley Ivan Hurt, Nov 02 2017

a(10) = 3, a(100) = 30, a(10^3) = 302, a(10^4) = 3041, a(10^5) = 30393, a(10^6) = 303968, a(10^7) = 3039658, a(10^8) = 30396350, a(10^9) = 303963598, a(10^10) = 3039635373, a(10^11) = 30396355273, a(10^12) = 303963551068, a(10^13) = 3039635509338, a(10^14) = 30396355094469, a(10^15) = 303963550926043, a(10^16) = 3039635509271763, a(10^17) = 30396355092700721, and a(10^18) = 303963550927014110. The limit of a(n)/n is 3/Pi^2. - Charles R Greathouse IV, Nov 04 2017

			a(4)=2; there are two partitions of 4 into two parts: (3,1) and (2,2). Both of the larger parts are squarefree, thus a(4)=2.
a(5)=1; there are two partitions of 5 into two parts: (4,1) and (3,2). Among the larger parts, only 3 is squarefree, thus a(5)=1.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 150 terms from G. C. Greubel)
Index entries for sequences related to partitions

Cf. A008683, A071068, A261985, A262351, A262869, A262870, A262871, A262991, A262992, A013928.

with(numtheory): A262868:=n->add(mobius(n-i)^2, i=1..floor(n/2)): seq(A262868(n), n=1..100);

Table[Sum[MoebiusMu[n - i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,1]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Jan 03 2022 *)

a(n) = sum(i=1, n\2, moebius(n-i)^2); \\ Michel Marcus, Oct 04 2015

f(n)=my(s); forfactored(k=1,sqrtint(n),s+=n\k[1]^2*moebius(k)); s
a(n)=n--; f(n) - f(n\2) \\ Charles R Greathouse IV, Nov 04 2017

a(n) = Sum_{i=1..floor(n/2)} mu(n-i)^2, where mu is the Möbius function A008683.
a(n) = A262991(n) - A262869(n).
a(n) ~ 3*n/Pi^2. - Charles R Greathouse IV, Nov 04 2017

A262869 Number of squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

Original entry on oeis.org

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

Offset: 1

Wesley Ivan Hurt, Oct 03 2015

Number of distinct rectangles with integer length and squarefree width such that L + W = n, W <= L. For example, a(14) = 6; the rectangles are 13 X 1, 12 X 2, 11 X 3, 9 X 5, 8 X 6, 7 X 7. - Wesley Ivan Hurt, Nov 04 2017

			a(5)=2; there are two partitions of 5 into two parts: (4,1) and (3,2). Both of the smaller parts are squarefree, thus a(5)=2.
a(6)=3; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). Among the three smaller parts, all are squarefree, thus a(6)=3.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A008683, A013928, A071068, A261985, A262351, A262868, A262870, A262871, A262991, A262992.

with(numtheory): A262869:=n->add(mobius(i)^2, i=1..floor(n/2)): seq(A262869(n), n=1..100);

Table[Sum[MoebiusMu[i]^2, {i, Floor[n/2]}], {n, 100}]
Table[Count[IntegerPartitions[n,{2}][[All,2]],?SquareFreeQ],{n,80}] (* _Harvey P. Dale, Oct 17 2021 *)

a(n) = sum(i=1, n\2, moebius(i)^2); \\ Michel Marcus, Oct 04 2015

a(n)=my(s); n\=2; forsquarefree(k=1, sqrtint(n), s += n\k[1]^2*moebius(k)); s \\ Charles R Greathouse IV, Jan 08 2018

a(n) = Sum_{i=1..floor(n/2)} mu(i)^2, where mu is the Möebius function (A008683).
a(n) = A262991(n) - A262868(n).
a(n) = A013928(floor(n/2)+1). - Georg Fischer, Nov 29 2022

A262871 Sum of the squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

Original entry on oeis.org

0, 1, 1, 3, 3, 6, 6, 6, 6, 11, 11, 17, 17, 24, 24, 24, 24, 24, 24, 34, 34, 45, 45, 45, 45, 58, 58, 72, 72, 87, 87, 87, 87, 104, 104, 104, 104, 123, 123, 123, 123, 144, 144, 166, 166, 189, 189, 189, 189, 189, 189, 215, 215, 215, 215, 215, 215, 244, 244, 274

Offset: 1

Wesley Ivan Hurt, Oct 03 2015

			a(5)=3; there are two partitions of 5 into two parts: (4,1) and (3,2). The sum of the smaller squarefree parts is 1+2=3. Thus a(5)=3.
a(6)=6; there are three partitions of 6 into two parts: (5,1), (4,2) and (3,3). All of the smaller parts are squarefree, so a(6) = 1+2+3 = 6.

Index entries for sequences related to partitions

Cf. A008683, A071068, A261985, A262351, A262868, A262869, A262870, A262991, A262992.

with(numtheory): A262871:=n->add(i*mobius(i)^2, i=1..floor(n/2)): seq(A262871(n), n=1..100);

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

a(n) = sum(i=1, n\2, i * moebius(i)^2); \\ Michel Marcus, Oct 04 2015

a(n)=my(s); forsquarefree(k=1,n\2, s += k[1]); s \\ Charles R Greathouse IV, Jan 08 2018

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

a(n) = A262992(n) - A262870(n).

A262992 Sum of the squarefree numbers among the partition parts of n into two parts.

Original entry on oeis.org

0, 2, 3, 8, 6, 14, 17, 24, 24, 29, 34, 51, 45, 65, 72, 87, 87, 104, 104, 133, 123, 155, 166, 189, 189, 202, 215, 229, 215, 259, 274, 305, 305, 355, 372, 407, 407, 463, 482, 521, 521, 583, 604, 669, 647, 670, 693, 740, 740, 740, 740, 817, 791, 844, 844, 899

Offset: 1

Wesley Ivan Hurt, Oct 06 2015

			a(3)=3; there is one partition of 3 into two parts: (2,1). The sum of the squarefree parts of this partition is 2+1=3, so a(3)=3.
a(5)=6; there are two partitions of 5 into two parts: (4,1) and (3,2). The sum of the squarefree parts of these partitions is 3+2+1=6, so a(5)=6.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to partitions

Cf. A066779, A008683, A071068, A261985, A262351, A262868, A262870, A262871, A262991.

with(numtheory): A262992:=n->add(i*mobius(i)^2 + (n-i)*mobius(n-i)^2, i=1..floor(n/2)): seq(A262992(n), n=1..100);

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

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

a(n)=my(s, k2, m=n-1); forsquarefree(k=1, sqrtint(m), k2=k[1]^2; s+= k2*binomial(m\k2+1, 2)*moebius(k)); s + (n%4==2 && issquarefree(n/2))*n/2 \\ Charles R Greathouse IV, Jan 13 2018

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

a(n) = A262870(n) + A262871(n).

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

Original entry on oeis.org

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

Wesley Ivan Hurt, Oct 26 2017

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

			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

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

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

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 *)

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

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

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

cp's OEIS Frontend

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A262868 Number of squarefree numbers appearing among the larger parts of the partitions of n into two parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A262869 Number of squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A262871 Sum of the squarefree numbers appearing among the smaller parts of the partitions of n into two parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

A262992 Sum of the squarefree numbers among the partition parts of n into two parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

PARI

Formula