A308952 -id:A308952 - cp's OEIS frontend

A308953 Sum of all the parts in the partitions of n into 7 squarefree parts.

0, 0, 0, 0, 0, 0, 0, 7, 8, 18, 20, 44, 60, 104, 126, 180, 224, 340, 396, 551, 640, 882, 1034, 1357, 1536, 2025, 2314, 2943, 3304, 4176, 4680, 5797, 6464, 7887, 8806, 10605, 11664, 14023, 15504, 18291, 20040, 23657, 25956, 30272, 32956, 38295, 41860, 48269

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308954, A308955, A308956, A308957, A308958, A308959, A308960.

Table[Total[Flatten[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&]]],{n,0,50}] (* Harvey P. Dale, Feb 25 2024 *)

a(n) = n * Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2, where mu is the Möbius function (A008683).
a(n) = n * A308952(n).
a(n) = A308954(n) + A308955(n) + A308956(n) + A308957(n) + A308958(n) + A308959(n) + A308960(n).

A308954 Sum of the smallest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 8, 10, 13, 15, 22, 25, 33, 37, 49, 55, 71, 77, 98, 109, 136, 148, 182, 199, 243, 264, 314, 344, 413, 441, 522, 567, 663, 711, 829, 896, 1036, 1106, 1269, 1370, 1572, 1666, 1903, 2041, 2316, 2460, 2780, 2971, 3350, 3546

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308955, A308956, A308957, A308958, A308959, A308960.

Table[Sum[Sum[Sum[Sum[Sum[Sum[o * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * o, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308955(n) - A308956(n) - A308957(n) - A308958(n) - A308959(n) - A308960(n).

A308955 Sum of the sixth largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 5, 9, 11, 14, 17, 25, 29, 39, 43, 58, 67, 85, 93, 120, 136, 168, 185, 229, 255, 311, 342, 414, 459, 547, 593, 711, 782, 911, 987, 1159, 1270, 1467, 1580, 1823, 1990, 2276, 2441, 2799, 3035, 3435, 3686, 4177, 4505, 5062

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308954, A308956, A308957, A308958, A308959, A308960.

Table[Sum[Sum[Sum[Sum[Sum[Sum[m * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&][[;;,6]]],{n,0,60}] (* Harvey P. Dale, Mar 07 2025 *)

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * m, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308956(n) - A308957(n) - A308958(n) - A308959(n) - A308960(n).

A308956 Sum of the fifth largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 4, 6, 10, 12, 16, 20, 30, 34, 46, 52, 71, 80, 103, 115, 149, 167, 211, 236, 298, 332, 410, 457, 559, 619, 742, 814, 981, 1078, 1269, 1384, 1633, 1786, 2072, 2246, 2607, 2839, 3267, 3524, 4050, 4379, 4970, 5350, 6076, 6555

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308954, A308955, A308957, A308958, A308959, A308960.

Table[Sum[Sum[Sum[Sum[Sum[Sum[l * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * l, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308955(n) - A308957(n) - A308958(n) - A308959(n) - A308960(n).

A308957 Sum of the fourth largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 2, 5, 7, 11, 14, 20, 24, 36, 41, 56, 64, 86, 98, 129, 147, 193, 222, 284, 324, 409, 457, 567, 635, 773, 862, 1037, 1147, 1375, 1516, 1778, 1953, 2290, 2510, 2920, 3186, 3680, 4017, 4614, 4996, 5734, 6226, 7081, 7682, 8732, 9450

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308954, A308955, A308956, A308958, A308959, A308960.

Table[Sum[Sum[Sum[Sum[Sum[Sum[k * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]
Table[Total[Select[IntegerPartitions[n,{7}],AllTrue[#,SquareFreeQ]&][[;;,4]]],{n,0,60}] (* Harvey P. Dale, Sep 24 2023 *)

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * k, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308958(n) - A308959(n) - A308960(n).

A308958 Sum of the third largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 6, 8, 14, 17, 25, 30, 44, 50, 72, 83, 115, 136, 184, 213, 278, 321, 409, 463, 579, 650, 807, 900, 1089, 1215, 1462, 1610, 1926, 2133, 2520, 2772, 3258, 3586, 4195, 4587, 5327, 5847, 6780, 7376, 8513, 9283, 10639, 11538, 13168

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308954, A308955, A308956, A308957, A308959, A308960.

Table[Sum[Sum[Sum[Sum[Sum[Sum[j * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * j, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308957(n) - A308959(n) - A308960(n).

A308959 Sum of the second largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 4, 8, 10, 18, 22, 34, 41, 65, 78, 111, 129, 179, 213, 277, 315, 412, 476, 602, 674, 849, 966, 1197, 1341, 1645, 1857, 2251, 2496, 3014, 3361, 3995, 4391, 5205, 5731, 6722, 7323, 8538, 9331, 10791, 11678, 13468, 14616, 16687

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308954, A308955, A308956, A308957, A308958, A308960.

Table[Sum[Sum[Sum[Sum[Sum[Sum[i * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * i, where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308957(n) - A308958(n) - A308960(n).

A308960 Sum of the largest parts in the partitions of n into 7 squarefree parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 5, 13, 19, 34, 40, 58, 77, 118, 139, 194, 232, 324, 385, 508, 576, 775, 883, 1133, 1274, 1630, 1821, 2262, 2525, 3093, 3450, 4153, 4563, 5494, 6067, 7155, 7842, 9283, 10177, 11860, 12928, 15051, 16466, 18969, 20543, 23705, 25820

Offset: 0

Wesley Ivan Hurt, Jul 03 2019

nonn

Index entries for sequences related to partitions

Cf. A008683, A308952, A308953, A308954, A308955, A308956, A308957, A308958, A308959.

Table[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o) * MoebiusMu[o]^2 * MoebiusMu[m]^2 * MoebiusMu[l]^2 * MoebiusMu[k]^2 * MoebiusMu[j]^2 * MoebiusMu[i]^2 * MoebiusMu[n - i - j - k - l - m - o]^2, {i, j, Floor[(n - j - k - l - m - o)/2]}], {j, k, Floor[(n - k - l - m - o)/3]}], {k, l, Floor[(n - l - m - o)/4]}], {l, m, Floor[(n - m - o)/5]}], {m, o, Floor[(n - o)/6]}], {o, Floor[n/7]}], {n, 0, 50}]

a(n) = Sum_{o=1..floor(n/7)} Sum_{m=o..floor((n-o)/6)} Sum_{l=m..floor((n-m-o)/5)} Sum_{k=l..floor((n-l-m-o)/4)} Sum_{j=k..floor((n-k-l-m-o)/3)} Sum_{i=j..floor((n-j-k-l-m-o)/2)} mu(o)^2 * mu(m)^2 * mu(l)^2 * mu(k)^2 * mu(j)^2 * mu(i)^2 * mu(n-i-j-k-l-m-o)^2 * (n-i-j-k-l-m-o), where mu is the Möbius function (A008683).

a(n) = A308953(n) - A308954(n) - A308955(n) - A308956(n) - A308957(n) - A308958(n) - A308959(n).

A341067 Number of ways to write n as an ordered sum of 7 squarefree numbers.

Original entry on oeis.org

1, 7, 28, 77, 168, 315, 553, 932, 1505, 2282, 3297, 4634, 6447, 8771, 11607, 15029, 19390, 24885, 31500, 39137, 48335, 59584, 73003, 88109, 105525, 126112, 150472, 177632, 208160, 243194, 284102, 329357, 379379, 435477, 500108, 571124, 648998, 735112, 833483, 940765, 1057679

Offset: 7

Ilya Gutkovskiy, Feb 04 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..10000

Cf. A005117, A008683, A008966, A098235, A280210, A308952, A341064, A341065, A341066, A341068, A341069, A341070.

b:= proc(n, t) option remember;
      `if`(n=0, `if`(t=0, 1, 0), `if`(t<1, 0, add(
      `if`(numtheory[issqrfree](j), b(n-j, t-1), 0), j=1..n)))
    end:
a:= n-> b(n, 7):
seq(a(n), n=7..47);  # Alois P. Heinz, Feb 04 2021

nmax = 47; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^k, {k, 1, nmax}]^7, {x, 0, nmax}], x] // Drop[#, 7] &

G.f.: (Sum_{k>=1} mu(k)^2 * x^k)^7.

A341095 Number of partitions of n into 7 distinct squarefree parts.

Original entry on oeis.org

1, 1, 0, 1, 2, 2, 2, 4, 6, 7, 7, 10, 14, 15, 15, 21, 28, 32, 32, 44, 53, 60, 60, 76, 93, 103, 107, 131, 157, 172, 178, 211, 247, 273, 283, 333, 384, 423, 439, 507, 577, 629, 657, 747, 846, 917, 960, 1078, 1211, 1306, 1362, 1521, 1691, 1822, 1898, 2103, 2322, 2494, 2596, 2850, 3134

Offset: 34

Ilya Gutkovskiy, Feb 04 2021

nonn

Cf. A005117, A008966, A098236, A307835, A308952, A341067, A341073, A341074, A341075, A341096, A341097, A341098.

b:= proc(n, i, t) option remember; `if`(n=0,
      `if`(t=0, 1, 0), `if`(i<1 or t<1, 0, b(n, i-1, t)+
      `if`(numtheory[issqrfree](i), b(n-i, min(n-i, i-1), t-1), 0)))
    end:
a:= n-> b(n$2, 7):
seq(a(n), n=34..94);  # Alois P. Heinz, Feb 04 2021

b[n_, i_, t_] := b[n, i, t] = If[n == 0,
     If[t == 0, 1, 0], If[i < 1 || t < 1, 0, b[n, i - 1, t] +
     If[SquareFreeQ[i], b[n - i, Min[n - i, i - 1], t - 1], 0]]];
a[n_] := b[n, n, 7];
Table[a[n], {n, 34, 94}] (* Jean-François Alcover, Feb 24 2022, after Alois P. Heinz *)

cp's OEIS Frontend

A308953 Sum of all the parts in the partitions of n into 7 squarefree parts.

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A308954 Sum of the smallest parts in the partitions of n into 7 squarefree parts.

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A308955 Sum of the sixth largest parts in the partitions of n into 7 squarefree parts.

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A308956 Sum of the fifth largest parts in the partitions of n into 7 squarefree parts.

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A308957 Sum of the fourth largest parts in the partitions of n into 7 squarefree parts.

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A308958 Sum of the third largest parts in the partitions of n into 7 squarefree parts.

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A308959 Sum of the second largest parts in the partitions of n into 7 squarefree parts.

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A308960 Sum of the largest parts in the partitions of n into 7 squarefree parts.

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A341067 Number of ways to write n as an ordered sum of 7 squarefree numbers.

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