cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A308952 Number of 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, 9, 12, 14, 20, 22, 29, 32, 42, 47, 59, 64, 81, 89, 109, 118, 144, 156, 187, 202, 239, 259, 303, 324, 379, 408, 469, 501, 577, 618, 704, 749, 851, 910, 1027, 1088, 1228, 1308, 1461, 1548, 1730, 1838, 2039, 2153, 2387
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

Cf. A008683, A308953, A308954, A308955, A308956, A308957, A308958, A308959, A308960.

Programs

  • Maple
    g:= proc(n,k,m) option remember; local i , j;
 if m=1 then if n=k then return 1 else return 0 fi fi;
 if k*m < n then return 0 fi;
 if k*m = n then return 1 fi;
 add(add(procname(n-i*k,j,m-i),  j= select(numtheory:-issqrfree,[$max(1,ceil((n-i*k)/(m-i))) .. k-1])), i=1..min(n/k,m-1));
end proc:
f:= proc(n) local k;
  add(g(n,k,7),k=select(numtheory:-issqrfree,[$ceil(n/7)..n]))
end proc:
f(0):= 0:
map(f, [$0..100]); # Robert Israel, Jul 03 2019
  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[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}]

Formula

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, where mu is the Möbius function (A008683).
a(n) = A308953(n)/n for n > 0.

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

Original entry on oeis.org

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

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

Formula

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

Formula

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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).

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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

Views

Author

Wesley Ivan Hurt, Jul 03 2019

Keywords

Links

Crossrefs

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

Programs

  • Mathematica
    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}]

Formula

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).
Showing 1-8 of 8 results.

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