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 1-6 of 6 results.

A309466 Sum of the prime parts in the partitions of n into 5 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 67, 88, 136, 169, 248, 295, 413, 496, 652, 772, 1001, 1161, 1469, 1697, 2096, 2398, 2923, 3316, 3975, 4501, 5302, 5955, 6953, 7757, 8994, 9988, 11450, 12674, 14427, 15883, 17992, 19741, 22176, 24268, 27149, 29569, 32919
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Examples

			The partitions of n into 5 parts for n = 10, 11, ..
                                                       1+1+1+1+10
                                                        1+1+1+2+9
                                                        1+1+1+3+8
                                                        1+1+1+4+7
                                                        1+1+1+5+6
                                            1+1+1+1+9   1+1+2+2+8
                                            1+1+1+2+8   1+1+2+3+7
                                            1+1+1+3+7   1+1+2+4+6
                                            1+1+1+4+6   1+1+2+5+5
                                            1+1+1+5+5   1+1+3+3+6
                                1+1+1+1+8   1+1+2+2+7   1+1+3+4+5
                                1+1+1+2+7   1+1+2+3+6   1+1+4+4+4
                                1+1+1+3+6   1+1+2+4+5   1+2+2+2+7
                    1+1+1+1+7   1+1+1+4+5   1+1+3+3+5   1+2+2+3+6
                    1+1+1+2+6   1+1+2+2+6   1+1+3+4+4   1+2+2+4+5
                    1+1+1+3+5   1+1+2+3+5   1+2+2+2+6   1+2+3+3+5
        1+1+1+1+6   1+1+1+4+4   1+1+2+4+4   1+2+2+3+5   1+2+3+4+4
        1+1+1+2+5   1+1+2+2+5   1+1+3+3+4   1+2+2+4+4   1+3+3+3+4
        1+1+1+3+4   1+1+2+3+4   1+2+2+2+5   1+2+3+3+4   2+2+2+2+6
        1+1+2+2+4   1+1+3+3+3   1+2+2+3+4   1+3+3+3+3   2+2+2+3+5
        1+1+2+3+3   1+2+2+2+4   1+2+3+3+3   2+2+2+2+5   2+2+2+4+4
        1+2+2+2+3   1+2+2+3+3   2+2+2+2+4   2+2+2+3+4   2+2+3+3+4
        2+2+2+2+2   2+2+2+2+3   2+2+2+3+3   2+2+3+3+3   2+3+3+3+3
--------------------------------------------------------------------------
  n  |     10          11          12          13          14        ...
--------------------------------------------------------------------------
a(n) |     41          67          88         136         169        ...
--------------------------------------------------------------------------
- _Wesley Ivan Hurt_, Sep 12 2019

Links

Crossrefs

Cf. A010051, A309427, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

  • Mathematica
    Table[Total[Select[Flatten[IntegerPartitions[n,{5}]],PrimeQ]],{n,0,50}] (* Harvey P. Dale, Dec 31 2021 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-1)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} (i * c(i) + j * c(j) + k * c(k) + l * c(l) + (n-i-j-k-l) * c(n-i-j-k-l)), where c is the prime characteristic (A010051).

A309467 Sum of the prime parts in the partitions of n into 6 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 79, 101, 160, 210, 310, 392, 559, 683, 909, 1126, 1464, 1766, 2250, 2687, 3345, 3977, 4853, 5701, 6886, 8012, 9522, 11036, 12979, 14888, 17388, 19842, 22936, 26053, 29853, 33725, 38496, 43219, 48947, 54800, 61768, 68800
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A010051, A309433, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + l (PrimePi[l] - PrimePi[l - 1]) + m (PrimePi[m] - PrimePi[m - 1]) + (n - i - j - k - l - m) (PrimePi[n - i - j - k - l - m] - PrimePi[n - i - j - k - l - m - 1]), {i, j, Floor[(n - j - k - l - m)/2]}], {j, k, Floor[(n - k - l - m)/3]}], {k, l, Floor[(n - l - m)/4]}], {l, m, Floor[(n - m)/5]}], {m, Floor[n/6]}], {n, 0, 80}]

Formula

a(n) = Sum_{m=1..floor(n/6)} Sum_{l=m..floor((n-m)/5)} Sum_{k=l..floor((n-l-m)/4)} Sum_{j=k..floor((n-k-l-m)/3)} Sum_{i=j..floor((n-j-k-l-m)/2)} (i * c(i) + j * c(j) + k * c(k) + l * c(l) + m * c(m) + (n-i-j-k-l-m) * c(n-i-j-k-l-m)), where c is the prime characteristic (A010051).

A309468 Sum of the prime parts in the partitions of n into 7 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 79, 115, 175, 238, 357, 464, 670, 851, 1145, 1441, 1908, 2349, 3034, 3698, 4657, 5635, 7007, 8350, 10240, 12124, 14609, 17192, 20549, 23920, 28326, 32802, 38437, 44287, 51520, 58934, 68170, 77621, 89049, 100981
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A010051, A309436, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + l (PrimePi[l] - PrimePi[l - 1]) + m (PrimePi[m] - PrimePi[m - 1]) + o (PrimePi[o] - PrimePi[o - 1]) + (n - i - j - k - l - m - o) (PrimePi[n - i - j - k - l - m - o] - PrimePi[n - i - j - k - l - m - o - 1]), {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, 80}]

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)} (i * c(i) + j * c(j) + k * c(k) + l * c(l) + m * c(m) + o * c(o) + (n-i-j-k-l-m-o) * c(n-i-j-k-l-m-o)), where c = A010051.

A309469 Sum of the prime parts in the partitions of n into 8 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 79, 115, 191, 255, 389, 517, 752, 976, 1335, 1707, 2289, 2870, 3737, 4639, 5904, 7246, 9088, 11040, 13635, 16416, 19984, 23856, 28776, 34054, 40667, 47796, 56553, 66043, 77527, 89992, 104963, 121151, 140303
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A010051, A309437, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + l (PrimePi[l] - PrimePi[l - 1]) + m (PrimePi[m] - PrimePi[m - 1]) + o (PrimePi[o] - PrimePi[o - 1]) + p (PrimePi[p] - PrimePi[p - 1]) + (n - i - j - k - l - m - o - p) (PrimePi[n - i - j - k - l - m - o - p] - PrimePi[n - i - j - k - l - m - o - p - 1]), {i, j, Floor[(n - j - k - l - m - o - p)/2]}], {j, k, Floor[(n - k - l - m - o - p)/3]}], {k, l, Floor[(n - l - m - o - p)/4]}], {l, m, Floor[(n - m - o - p)/5]}], {m, o, Floor[(n - o - p)/6]}], {o, p, Floor[(n - p)/7]}], {p, Floor[n/8]}], {n, 0, 50}]

Formula

a(n) = Sum_{p=1..floor(n/8)} Sum_{o=p..floor((n-p)/7)} Sum_{m=o..floor((n-o-p)/6)} Sum_{l=m..floor((n-m-o-p)/5)} Sum_{k=l..floor((n-l-m-o-p)/4)} Sum_{j=k..floor((n-k-l-m-o-p)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p)/2)} (i * c(i) + j * c(j) + k * c(k) + l * c(l) + m * c(m) + o * c(o) + p * c(p) + (n-i-j-k-l-m-o-p) * c(n-i-j-k-l-m-o-p)), where c is the prime characteristic (A010051).

A309470 Sum of the prime parts in the partitions of n into 9 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 79, 115, 191, 273, 408, 553, 811, 1068, 1474, 1919, 2585, 3295, 4343, 5446, 6999, 8701, 10998, 13509, 16863, 20502, 25220, 30430, 37036, 44292, 53431, 63414, 75762, 89378, 105914, 124117, 146169, 170271, 199086
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A010051, A309438, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + l (PrimePi[l] - PrimePi[l - 1]) + m (PrimePi[m] - PrimePi[m - 1]) + o (PrimePi[o] - PrimePi[o - 1]) + p (PrimePi[p] - PrimePi[p - 1]) + q (PrimePi[q] - PrimePi[q - 1]) + (n - i - j - k - l - m - o - p - q) (PrimePi[n - i - j - k - l - m - o - p - q] - PrimePi[n - i - j - k - l - m - o - p - q - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q)/2]}], {j, k, Floor[(n - k - l - m - o - p - q)/3]}], {k, l, Floor[(n - l - m - o - p - q)/4]}], {l, m, Floor[(n - m - o - p - q)/5]}], {m, o, Floor[(n - o - p - q)/6]}], {o, p, Floor[(n - p - q)/7]}], {p, q, Floor[(n - q)/8]}], {q, Floor[n/9]}], {n, 0, 50}]

Formula

a(n) = Sum_{q=1..floor(n/9)} Sum_{p=q..floor((n-q)/8)} Sum_{o=p..floor((n-p-q)/7)} Sum_{m=o..floor((n-o-p-q)/6)} Sum_{l=m..floor((n-m-o-p-q)/5)} Sum_{k=l..floor((n-l-m-o-p-q)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q)/2)} (q * c(q) + p * c(p) + o * c(o) + m * c(m) + l * c(l) + k * c(k) + j * c(j) + i * c(i) + (n-i-j-k-l-m-o-p-q) * c(n-i-j-k-l-m-o-p-q)), where c is the prime characteristic (A010051).

A309471 Sum of the prime parts in the partitions of n into 10 parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 7, 11, 28, 41, 79, 115, 191, 273, 428, 574, 851, 1133, 1576, 2072, 2819, 3621, 4812, 6112, 7918, 9931, 12655, 15684, 19714, 24221, 29987, 36534, 44796, 54051, 65660, 78684, 94653, 112671, 134499, 159012, 188569, 221650
Offset: 0

Views

Author

Wesley Ivan Hurt, Aug 03 2019

Keywords

Links

Crossrefs

Cf. A010051, A309439, A309465, A309466, A309467, A309468, A309469, A309470, A309471.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i (PrimePi[i] - PrimePi[i - 1]) + j (PrimePi[j] - PrimePi[j - 1]) + k (PrimePi[k] - PrimePi[k - 1]) + l (PrimePi[l] - PrimePi[l - 1]) + m (PrimePi[m] - PrimePi[m - 1]) + o (PrimePi[o] - PrimePi[o - 1]) + p (PrimePi[p] - PrimePi[p - 1]) + q (PrimePi[q] - PrimePi[q - 1]) + r (PrimePi[r] - PrimePi[r - 1]) + (n - i - j - k - l - m - o - p - q - r) (PrimePi[n - i - j - k - l - m - o - p - q - r] - PrimePi[n - i - j - k - l - m - o - p - q - r - 1]), {i, j, Floor[(n - j - k - l - m - o - p - q - r)/2]}], {j, k, Floor[(n - k - l - m - o - p - q - r)/3]}], {k, l, Floor[(n - l - m - o - p - q - r)/4]}], {l, m, Floor[(n - m - o - p - q - r)/5]}], {m, o, Floor[(n - o - p - q - r)/6]}], {o, p, Floor[(n - p - q - r)/7]}], {p, q, Floor[(n - q - r)/8]}], {q, r, Floor[(n - r)/9]}], {r, Floor[n/10]}], {n, 0, 80}]

Formula

a(n) = Sum_{r=1..floor(n/10)} Sum_{q=r..floor((n-r)/9)} Sum_{p=q..floor((n-q-r)/8)} Sum_{o=p..floor((n-p-q-r)/7)} Sum_{m=o..floor((n-o-p-q-r)/6)} Sum_{l=m..floor((n-m-o-p-q-r)/5)} Sum_{k=l..floor((n-l-m-o-p-q-r)/4)} Sum_{j=k..floor((n-k-l-m-o-p-q-r)/3)} Sum_{i=j..floor((n-j-k-l-m-o-p-q-r)/2)} (r * c(r) + q * c(q) + p * c(p) + o * c(o) + m * c(m) + l * c(l) + k * c(k) + j * c(j) + i * c(i) + (n-i-j-k-l-m-o-p-q-r) * c(n-i-j-k-l-m-o-p-q-r)), where c is the prime characteristic (A010051).
Showing 1-6 of 6 results.

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