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 1-9 of 9 results.

A259198 Number of partitions of n into eight primes.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 4, 5, 6, 7, 8, 10, 9, 12, 14, 16, 16, 21, 19, 26, 26, 31, 30, 39, 34, 46, 43, 53, 48, 65, 56, 77, 66, 85, 77, 104, 84, 118, 99, 133, 112, 155, 123, 177, 143, 196, 162, 227, 174, 256, 200, 282, 220, 318, 241, 360, 270, 389, 300, 442, 322
Offset: 16

Views

Author

Doug Bell, Jun 20 2015

Keywords

Examples

			a(20) = 2 because there are 2 partitions of 20 into eight primes: [2,2,2,2,2,2,3,5] and [2,2,2,2,3,3,3,3].

Links

Crossrefs

Column k=8 of A117278.
Number of partitions of n into r primes for r = 1-10: A010051, A061358, A068307, A259194, A259195, A259196, A259197, this sequence, A259200, A259201.
Cf. A000040.

Programs

  • Mathematica
    Table[Length@IntegerPartitions[n, {8}, Prime@Range@100], {n, 16, 100}] (* Robert Price, Apr 25 2025 *)

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)} A010051(i) * A010051(j) * A010051(k) * A010051(l) * A010051(m) * A010051(o) * A010051(p) * A010051(n-i-j-k-l-m-o-p). - Wesley Ivan Hurt, Apr 17 2019
a(n) = [x^n y^8] Product_{k>=1} 1/(1 - y*x^prime(k)). - Ilya Gutkovskiy, Apr 18 2019
a(n) = A326455(n)/n for n > 0. - Wesley Ivan Hurt, Jul 06 2019

A326456 Sum of the smallest parts of the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 8, 11, 12, 15, 16, 22, 18, 26, 28, 36, 32, 47, 38, 59, 52, 71, 60, 93, 68, 109, 86, 128, 96, 157, 112, 190, 132, 210, 154, 262, 168, 300, 198, 344, 224, 399, 246, 464, 286, 515, 324, 605
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326457, A326458, A326459, A326460, A326461, A326462, A326463.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{8}],AllTrue[#,PrimeQ]&][[;;,-1]]],{n,0,70}] (* Harvey P. Dale, Mar 28 2025 *)

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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * p, where c = A010051.
a(n) = A326455(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

A326457 Sum of the seventh largest parts in the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 8, 9, 11, 13, 15, 18, 22, 20, 26, 32, 36, 37, 47, 45, 59, 63, 73, 74, 95, 84, 111, 109, 132, 123, 163, 145, 196, 173, 220, 209, 278, 225, 316, 272, 366, 309, 427, 343, 494, 405, 557, 466, 659
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326458, A326459, A326460, A326461, A326462, A326463.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[o * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * o, where c = A010051.
a(n) = A326455(n) - A326456(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

A326458 Sum of the sixth largest parts of the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 6, 9, 9, 12, 13, 17, 18, 24, 20, 30, 32, 41, 37, 56, 47, 71, 65, 88, 76, 114, 88, 138, 115, 162, 129, 205, 157, 246, 187, 282, 225, 352, 247, 404, 298, 469, 339, 553, 385, 641, 453, 731, 522, 855
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326459, A326460, A326461, A326462, A326463.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[m * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * m, where c = A010051.
a(n) = A326455(n) - A326456(n) - A326457(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

A326459 Sum of the fifth largest parts of the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 7, 9, 10, 12, 15, 17, 20, 24, 24, 30, 39, 43, 47, 60, 58, 75, 81, 94, 97, 124, 116, 152, 146, 180, 172, 229, 207, 278, 246, 320, 298, 400, 332, 464, 395, 539, 463, 643, 521, 749, 612, 855, 720
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326460, A326461, A326462, A326463.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[l * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * l, where c = A010051.
a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326460(n) - A326461(n) - A326462(n) - A326463(n).

A326460 Sum of the fourth largest parts of the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 10, 10, 14, 15, 19, 20, 30, 26, 38, 43, 54, 51, 74, 64, 97, 93, 118, 111, 159, 132, 193, 172, 231, 202, 293, 243, 357, 296, 407, 352, 517, 402, 600, 495, 706, 577, 851, 661, 1004, 800, 1150
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326461, A326462, A326463.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[k * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * k, where c = A010051.
a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326461(n) - A326462(n) - A326463(n).

A326461 Sum of the third largest parts in the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 8, 10, 12, 14, 19, 21, 27, 34, 35, 44, 57, 64, 67, 88, 87, 115, 121, 142, 146, 191, 176, 233, 232, 289, 271, 369, 336, 455, 414, 537, 500, 687, 588, 816, 722, 974, 843, 1179, 977, 1392, 1172
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326460, A326462, A326463.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{8}],AllTrue[#,PrimeQ]&][[;;,3]]],{n,0,70}] (* Harvey P. Dale, Apr 13 2025 *)

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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * j, where c = A010051.
a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326462(n) - A326463(n).

A326462 Sum of the second largest parts in the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 6, 8, 14, 14, 19, 23, 31, 35, 46, 45, 62, 71, 86, 85, 119, 115, 162, 163, 205, 212, 275, 260, 356, 344, 435, 411, 559, 516, 709, 640, 829, 786, 1060, 914, 1272, 1112, 1485, 1299, 1795, 1501, 2133
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326460, A326461, A326463.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[i * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * i, where c = A010051.
a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326463(n).

A326463 Sum of the largest parts of the partitions of n into 8 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 8, 8, 15, 20, 20, 27, 40, 47, 62, 78, 73, 104, 132, 152, 172, 223, 211, 298, 324, 387, 394, 509, 470, 640, 645, 775, 756, 1015, 916, 1265, 1146, 1445, 1403, 1852, 1576, 2200, 1953, 2565, 2330, 3143
Offset: 0

Views

Author

Wesley Ivan Hurt, Jul 06 2019

Keywords

Links

Crossrefs

Cf. A010051, A259198, A326455, A326456, A326457, A326458, A326459, A326460, A326461, A326462.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[Sum[Sum[Sum[(n-i-j-k-l-m-o-p) * (PrimePi[i] - PrimePi[i - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[m] - PrimePi[m - 1]) (PrimePi[o] - PrimePi[o - 1]) (PrimePi[p] - PrimePi[p - 1]) (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)} c(p) * c(o) * c(m) * c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l-m-o-p) * (n-i-j-k-l-m-o-p), where c = A010051.
a(n) = A326455(n) - A326456(n) - A326457(n) - A326458(n) - A326459(n) - A326460(n) - A326461(n) - A326462(n).
Showing 1-9 of 9 results.

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