A280226 -id:A280226 - cp's OEIS frontend

A280250 Sum of the smaller parts of the partitions of 2n into 2 squarefree parts.

1, 3, 4, 6, 8, 14, 11, 17, 16, 32, 27, 39, 39, 58, 47, 61, 65, 93, 67, 95, 80, 130, 94, 142, 106, 203, 130, 189, 151, 232, 165, 246, 187, 311, 235, 362, 260, 389, 259, 377, 283, 442, 306, 473, 367, 511, 407, 530, 395, 625, 458, 673, 493, 801, 507, 782, 548, 842, 590, 901

Offset: 1

Wesley Ivan Hurt, Dec 29 2016

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

Cf. A008683, A185297, A280226, A280251, A280252.

with(numtheory): A280250:=n->sum(i*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280250(n), n=1..100);

Table[Total[Select[IntegerPartitions[2 n,{2}],AllTrue[#,SquareFreeQ]&][[;;,2]]],{n,60}] (* Harvey P. Dale, May 12 2025 *)

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

a(n) = A280252(n) - A280251(n).

A280251 Sum of the larger parts of the partitions of 2n into two squarefree parts.

Original entry on oeis.org

1, 5, 8, 18, 12, 34, 31, 63, 56, 88, 83, 129, 91, 138, 103, 195, 173, 303, 199, 345, 256, 442, 274, 482, 294, 525, 410, 539, 487, 668, 517, 714, 539, 913, 675, 1150, 776, 1131, 755, 1223, 783, 1406, 898, 1551, 1163, 1605, 1191, 1774, 1271, 1875, 1378, 2031, 1521, 2547

Offset: 1

Wesley Ivan Hurt, Dec 29 2016

Index entries for sequences related to partitions

Cf. A008683, A187129, A280226, A280250, A280252.

with(numtheory): A280251:=n->sum((2*n-i)*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280251(n), n=1..100);

Table[Total[Select[IntegerPartitions[2 n,{2}],AllTrue[#,SquareFreeQ]&][[;;,1]]],{n,60}] (* Harvey P. Dale, Apr 22 2023 *)

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

a(n) = A280252(n) - A280250(n).

A280252 Sum of the parts in the partitions of 2n into two squarefree parts.

Original entry on oeis.org

2, 8, 12, 24, 20, 48, 42, 80, 72, 120, 110, 168, 130, 196, 150, 256, 238, 396, 266, 440, 336, 572, 368, 624, 400, 728, 540, 728, 638, 900, 682, 960, 726, 1224, 910, 1512, 1036, 1520, 1014, 1600, 1066, 1848, 1204, 2024, 1530, 2116, 1598, 2304, 1666, 2500, 1836, 2704

Offset: 1

Wesley Ivan Hurt, Dec 29 2016

Index entries for sequences related to partitions

Cf. A008683, A226237, A280226, A280250, A280251.

with(numtheory): A280252:=n->2*n*add(mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280252(n), n=1..100);

Table[2 n*Sum[MoebiusMu[i]^2 MoebiusMu[2 n - i]^2, {i, n}], {n, 80}] (* Wesley Ivan Hurt, Jan 05 2024 *)

a(n) = 2*n * A280226(n).

a(n) = A280250(n) + A280251(n).

A280316 Sum of squares of parts of the partitions of 2n into two squarefree parts.

Original entry on oeis.org

2, 18, 44, 124, 108, 372, 398, 886, 888, 1560, 1642, 2778, 2098, 3440, 2810, 5618, 5350, 9766, 6934, 12382, 9744, 17448, 11112, 20440, 12728, 24050, 19508, 26610, 25270, 36108, 28950, 41020, 31974, 56038, 42490, 74484, 51668, 77210, 52810, 87970, 57074, 105804, 68972

Offset: 1

Wesley Ivan Hurt, Dec 31 2016

Index entries for sequences related to partitions

Cf. A008683, A280226, A280320, A280322.

with(numtheory): A280316:=n->add((i^2+(2*n-i)^2) * mobius(i)^2 * mobius(2*n-i)^2, i=1..n): seq(A280316(n), n=1..100);

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

a(n) = A280320(n) + A280322(n).

A280320 Sum of the squares of the smaller parts of the partitions of 2n into two squarefree parts.

Original entry on oeis.org

1, 5, 10, 14, 34, 66, 59, 75, 84, 220, 205, 309, 373, 600, 565, 665, 839, 1103, 959, 1191, 1176, 1860, 1416, 2060, 1664, 3653, 2194, 3505, 2891, 4974, 3563, 5534, 4371, 7551, 5845, 8874, 6742, 10409, 7061, 10145, 8037, 12414, 9030, 13327, 10849, 15319, 13473, 15960

Offset: 1

Wesley Ivan Hurt, Dec 31 2016

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

Cf. A008683, A280226, A280316, A280322.

with(numtheory): A280320:=n->add(i^2*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280320(n), n=1..100);

Table[Total[Select[IntegerPartitions[2 n,{2}],AllTrue[#,SquareFreeQ]&][[All,2]]^2],{n,50}] (* Harvey P. Dale, Jan 22 2023 *)

a(n) = sum(i=1, n, i^2*issquarefree(i)*issquarefree(2*n-i)); \\ Michel Marcus, May 16 2019

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

a(n) = A280316(n) - A280322(n).

A280322 Sum of the squares of the larger parts of the partitions of 2n into two squarefree parts.

Original entry on oeis.org

1, 13, 34, 110, 74, 306, 339, 811, 804, 1340, 1437, 2469, 1725, 2840, 2245, 4953, 4511, 8663, 5975, 11191, 8568, 15588, 9696, 18380, 11064, 20397, 17314, 23105, 22379, 31134, 25387, 35486, 27603, 48487, 36645, 65610, 44926, 66801, 45749, 77825, 49037, 93390, 59942

Offset: 1

Wesley Ivan Hurt, Dec 31 2016

Index entries for sequences related to partitions

Cf. A008683, A280226, A280316.

with(numtheory): A280322:=n->add((2*n-i)^2*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280322(n), n=1..100);

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

a(n) = A280316(n) - A280320(n).

A280634 Number of partitions of 2n into two refactorable parts.

Original entry on oeis.org

1, 1, 0, 0, 2, 0, 1, 1, 1, 2, 0, 1, 2, 0, 1, 1, 0, 2, 1, 0, 2, 1, 0, 3, 0, 1, 1, 0, 2, 1, 1, 2, 0, 2, 0, 2, 2, 1, 1, 3, 1, 2, 1, 1, 2, 3, 0, 5, 2, 2, 1, 2, 2, 3, 1, 4, 1, 4, 0, 5, 1, 2, 1, 3, 1, 3, 1, 3, 1, 5, 0, 7, 1, 3, 1, 3, 2, 3, 1, 5, 0, 6, 0, 7, 1, 3, 1, 5, 0, 3

Offset: 1

Wesley Ivan Hurt, Jan 06 2017

			a(5) = 2; There are two partitions of 2*5 = 10 into two refactorable parts: (1,9) and (2,8).

Eric Weisstein's World of Mathematics, Refactorable Number
Index entries for sequences related to partitions

Cf. A000005, A033950, A172398, A280226.

with(numtheory): A280634:=n->add((1-signum((i mod tau(i))))*(1-signum((2*n-i) mod tau(2*n-i))), i=1..n): seq(A280634(n), n=1..150);

Table[Sum[(1 - Sign[Mod[i, DivisorSigma[0, i]]]) (1 - Sign[Mod[#, DivisorSigma[0, #]]] &[2 n - i]), {i, n}], {n, 90}] (* Michael De Vlieger, Jan 07 2017 *)

a(n) = Sum_{i=1..n} (1-sign(i mod d(i))) * (1-sign((2n-i) mod d(2n-i))) where d(n) is the number of divisors of n.

A294248 Number of partitions of 2n into two distinct squarefree parts.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 2, 5, 4, 5, 4, 7, 4, 6, 4, 8, 6, 11, 6, 11, 7, 12, 7, 13, 8, 13, 10, 13, 10, 14, 10, 15, 10, 17, 12, 21, 13, 19, 12, 20, 12, 21, 13, 23, 17, 22, 16, 24, 17, 25, 17, 26, 18, 31, 18, 29, 19, 30, 19, 31, 19, 32, 23, 30, 22, 31, 22, 32, 22, 34

Offset: 1

Wesley Ivan Hurt, Oct 25 2017

Index entries for sequences related to partitions

Cf. A008683, A280226.

Table[Sum[MoebiusMu[i]^2*MoebiusMu[2 n - i]^2, {i, n - 1}], {n, 80}]

A294248(n)=sum(i=1,n-1,moebius(i)^2*moebius(2*n-i)^2); \\ R. J. Cano, Oct 25 2017

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

A280448 Sum of the GCDs of the smaller and larger parts of the partitions of 2n into two squarefree parts.

Original entry on oeis.org

1, 3, 4, 4, 6, 10, 9, 7, 6, 20, 15, 11, 17, 28, 19, 11, 23, 23, 25, 27, 36, 48, 30, 24, 12, 55, 16, 35, 39, 56, 41, 20, 55, 73, 55, 44, 50, 81, 65, 39, 53, 96, 56, 71, 33, 97, 63, 40, 29, 53, 88, 83, 71, 63, 91, 68, 98, 126, 78, 87, 80, 134, 65, 40, 107, 147, 89, 107, 119

Offset: 1

Wesley Ivan Hurt, Jan 03 2017

Index entries for sequences related to partitions

Cf. A008683, A234307, A280226.

with(numtheory): A280448:=n->add(gcd(2*n-i, i)*mobius(i)^2*mobius(2*n-i)^2, i=1..n): seq(A280448(n), n=1..100);

Table[Sum[GCD[k, 2*n - k]*MoebiusMu[k]^2 * MoebiusMu[2*n - k]^2, {k, 1,
n}], {n, 1, 50}] (* G. C. Greubel, Jan 05 2017 *)

for(n=1,50, print1(sum(k=1,n, gcd(k,2*n-k) * (moebius(k))^2 *(moebius(2*n-k))^2), ", ")) \\ G. C. Greubel, Jan 05 2017

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

A302391 Number of partitions of 2n into two parts with at least one nonsquarefree part.

Original entry on oeis.org

0, 0, 1, 1, 3, 2, 4, 3, 5, 4, 6, 5, 8, 7, 10, 8, 10, 7, 12, 9, 13, 9, 15, 11, 17, 12, 17, 15, 18, 15, 20, 17, 22, 16, 22, 15, 23, 18, 26, 20, 28, 20, 29, 21, 28, 23, 30, 24, 32, 25, 33, 26, 34, 23, 36, 27, 37, 27, 39, 29, 41, 29, 40, 34, 42, 34, 44, 36, 46

Offset: 1

Wesley Ivan Hurt, Apr 06 2018

nonn

Index entries for sequences related to partitions

Cf. A008683, A013929, A280226, A294097.

[&+[(1-MoebiusMu(2*n-k)^2*MoebiusMu(k)^2): k in [1..n]]: n in [1..70]]; // Vincenzo Librandi, Apr 09 2018

Table[Sum[1 - MoebiusMu[2 n - i]^2*MoebiusMu[i]^2, {i, n}], {n, 100}]

a(n) = sum(i=1, n, 1 - moebius(2*n-i)^2*moebius(i)^2); \\ Michel Marcus, Apr 09 2018

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

a(n) = n - A280226(n). - Wesley Ivan Hurt, Dec 11 2023

cp's OEIS Frontend

A280250 Sum of the smaller parts of the partitions of 2n into 2 squarefree parts.

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A280251 Sum of the larger parts of the partitions of 2n into two squarefree parts.

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A280252 Sum of the parts in the partitions of 2n into two squarefree parts.

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A280316 Sum of squares of parts of the partitions of 2n into two squarefree parts.

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A280320 Sum of the squares of the smaller parts of the partitions of 2n into two squarefree parts.

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A280322 Sum of the squares of the larger parts of the partitions of 2n into two squarefree parts.

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A280634 Number of partitions of 2n into two refactorable parts.

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A294248 Number of partitions of 2n into two distinct squarefree parts.

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