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.

Previous Showing 11-17 of 17 results.

A308854 Sum of all the parts in the partitions of n into 5 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 11, 12, 26, 28, 45, 48, 51, 54, 95, 80, 126, 132, 161, 144, 250, 182, 297, 252, 348, 330, 527, 352, 594, 442, 700, 504, 888, 570, 1053, 720, 1189, 882, 1505, 836, 1710, 1104, 1927, 1248, 2303, 1300, 2703, 1560, 2862, 1836
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Links

Crossrefs

Cf. A010051, A259195, A308855, A308856, A308857, A308858, A308859.

Programs

  • Mathematica
    Table[Total[Flatten[Select[IntegerPartitions[n,{5}],AllTrue[#,PrimeQ]&]]],{n,0,60}] (* Harvey P. Dale, Apr 18 2022 *)

Formula

a(n) = n * Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l), where c = A010051.
a(n) = n * A259195(n).
a(n) = A308855(n) + A308856(n) + A308857(n) + A308858(n) + A308859(n).

A308855 Sum of the smallest parts in the partitions of n into 5 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 4, 7, 6, 7, 6, 12, 8, 14, 12, 18, 12, 27, 14, 30, 18, 33, 22, 48, 22, 53, 26, 62, 28, 71, 30, 87, 36, 92, 42, 113, 38, 127, 48, 139, 52, 159, 52, 190, 60, 190, 68, 233, 66, 264, 76, 275, 82, 308, 82, 359, 90, 370
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Links

Crossrefs

Cf. A010051, A259195, A308854, A308856, A308857, A308858, A308859.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[l (PrimePi[l] - PrimePi[l - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k - l] - PrimePi[n - i - j - k - l - 1]), {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l) * l, where c = A010051.
a(n) = A308854(n) - A308856(n) - A308857(n) - A308858(n) - A308859(n).

A308856 Sum of the fourth largest parts in the partitions of n into 5 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 5, 7, 7, 7, 8, 12, 10, 14, 18, 20, 18, 29, 21, 32, 28, 37, 38, 56, 34, 59, 47, 72, 51, 85, 55, 101, 68, 112, 81, 139, 73, 151, 105, 179, 110, 209, 113, 244, 136, 258, 161, 323, 147, 354, 187, 387, 200, 436, 204, 501
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Links

Crossrefs

Cf. A010051, A259195, A308854, A308855, A308857, A308858, A308859.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[k (PrimePi[l] - PrimePi[l - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k - l] - PrimePi[n - i - j - k - l - 1]), {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l) * k, where c = A010051.
a(n) = A308854(n) - A308855(n) - A308857(n) - A308858(n) - A308859(n).

A308857 Sum of the third largest parts in the partitions of n into 5 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 8, 7, 9, 8, 16, 12, 21, 20, 27, 20, 41, 27, 46, 34, 54, 44, 78, 44, 86, 59, 104, 65, 129, 79, 152, 96, 175, 115, 221, 117, 249, 157, 295, 170, 342, 179, 394, 214, 430, 243, 517, 245, 584, 307, 643, 332, 730, 352
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Links

Crossrefs

Cf. A010051, A259195, A308854, A308855, A308856, A308858, A308859.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,PrimeQ]&][[All,3]]],{n,0,70}] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 06 2020 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l) * j, where c = A010051.
a(n) = A308854(n) - A308855(n) - A308856(n) - A308858(n) - A308859(n).

A308858 Sum of the second largest parts in the partitions of n into 5 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 6, 8, 11, 11, 13, 20, 20, 27, 28, 35, 32, 51, 41, 60, 53, 74, 75, 112, 83, 136, 103, 162, 126, 205, 143, 246, 170, 283, 219, 365, 217, 415, 276, 475, 310, 554, 320, 642, 376, 690, 446, 835, 443, 944, 532, 1019, 587
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Links

Crossrefs

Cf. A010051, A259195, A308854, A308855, A308856, A308857, A308859.

Programs

  • Mathematica
    Table[Total[Select[IntegerPartitions[n,{5}],AllTrue[#,PrimeQ]&][[All,2]]],{n,0,60}] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Mar 15 2020 *)

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l) * i, where c = A010051.
a(n) = A308854(n) - A308855(n) - A308856(n) - A308857(n) - A308859(n).

A308859 Sum of the largest parts in the partitions of n into 5 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 3, 3, 8, 8, 15, 17, 17, 19, 35, 30, 50, 54, 61, 62, 102, 79, 129, 119, 150, 151, 233, 169, 260, 207, 300, 234, 398, 263, 467, 350, 527, 425, 667, 391, 768, 518, 839, 606, 1039, 636, 1233, 774, 1294, 918, 1612, 947, 1844, 1102
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 28 2019

Keywords

Links

Crossrefs

Cf. A010051, A259195, A308854, A308855, A308856, A308857, A308858.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[Sum[(n-i-j-k-l) (PrimePi[l] - PrimePi[l - 1]) (PrimePi[k] - PrimePi[k - 1]) (PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k - l] - PrimePi[n - i - j - k - l - 1]), {i, j, Floor[(n - j - k - l)/2]}], {j, k, Floor[(n - k - l)/3]}], {k, l, Floor[(n - l)/4]}], {l, Floor[n/5]}], {n, 0, 50}]

Formula

a(n) = Sum_{l=1..floor(n/5)} Sum_{k=l..floor((n-l)/4)} Sum_{j=k..floor((n-k-l)/3)} Sum_{i=j..floor((n-j-k-l)/2)} c(l) * c(k) * c(j) * c(i) * c(n-i-j-k-l) * (n-i-j-k-l), where c = A010051.
a(n) = A308854(n) - A308855(n) - A308856(n) - A308857(n) - A308858(n).

A347609 Number of partitions of n into at most 5 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 5, 6, 6, 8, 8, 9, 10, 11, 11, 14, 14, 15, 17, 18, 19, 21, 22, 23, 25, 27, 27, 32, 29, 34, 33, 37, 37, 42, 39, 47, 44, 51, 47, 58, 50, 61, 57, 67, 61, 73, 65, 80, 71, 86, 75, 95, 79, 101, 86, 107, 92, 115, 95, 125, 103, 132, 108
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Crossrefs

Cf. A000040, A000607, A071335, A117278, A259195, A347552, A347578, A347610.

Formula

a(n) = Sum_{k=1..5} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021
Previous Showing 11-17 of 17 results.

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