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 1-4 of 4 results.

A308771 Sum of the smallest parts of the partitions of n into 4 prime parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 4, 5, 4, 7, 4, 8, 6, 10, 8, 18, 6, 18, 10, 21, 10, 28, 10, 38, 14, 34, 14, 47, 12, 51, 18, 55, 16, 68, 18, 81, 20, 73, 22, 105, 20, 110, 24, 113, 26, 136, 24, 161, 30, 147, 32, 187, 28, 200, 34, 204, 32, 237, 32, 262, 38, 246
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A010051, A259194, A308772, A308773, A308774, A308809.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[k (PrimePi[k] - PrimePi[k - 1])*(PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} c(k) * c(j) * c(i) * c(n-i-j-k) * k, where c = A010051.
a(n) = A308809(n) - A308772(n) - A308773(n) - A308774(n).

A308772 Sum of the third largest parts of the partitions of n into 4 prime parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 2, 5, 5, 5, 7, 6, 8, 10, 12, 13, 20, 11, 20, 20, 27, 18, 34, 21, 44, 28, 44, 31, 59, 30, 65, 46, 79, 41, 96, 49, 115, 58, 117, 64, 157, 64, 170, 73, 179, 80, 214, 80, 243, 98, 245, 114, 307, 106, 332, 124, 352, 124, 399, 124
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A010051, A259194, A308771, A308773, A308774, A308809.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[j (PrimePi[k] - PrimePi[k - 1])*(PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} c(k) * c(j) * c(i) * c(n-i-j-k) * j, where c = A010051.
a(n) = A308809(n) - A308771(n) - A308773(n) - A308774(n).

A308773 Sum of the second largest parts in the partitions of n into 4 prime parts.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 3, 5, 6, 5, 11, 8, 13, 12, 20, 17, 28, 15, 32, 26, 41, 24, 53, 33, 75, 48, 83, 57, 103, 54, 126, 80, 143, 71, 170, 93, 219, 112, 217, 122, 276, 120, 310, 145, 320, 148, 376, 160, 446, 190, 443, 218, 532, 196, 587, 240, 613, 246
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A010051, A259194, A308771, A308772, A308774, A308809.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[i (PrimePi[k] - PrimePi[k - 1])*(PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 100}]
Table[Total[Select[IntegerPartitions[n,{4}],AllTrue[#,PrimeQ]&][[;;,2]]],{n,0,70}] (* Harvey P. Dale, May 09 2025 *)

Formula

a(n) = Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} c(k) * c(j) * c(i) * c(n-i-j-k) * i, where c = A010051.
a(n) = A308809(n) - A308771(n) - A308772(n) - A308774(n).

A308809 Sum of all the parts in the partitions of n into 4 primes.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 8, 9, 10, 22, 24, 26, 42, 30, 48, 51, 72, 76, 120, 63, 132, 115, 168, 125, 234, 135, 308, 203, 330, 217, 416, 198, 476, 315, 540, 296, 684, 351, 840, 410, 798, 473, 1056, 450, 1196, 564, 1248, 637, 1500, 612, 1768, 795, 1782, 880
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 25 2019

Keywords

Links

Crossrefs

Cf. A010051, A259194, A308771, A308772, A308773, A308774.

Programs

  • Mathematica
    Table[n*Sum[Sum[Sum[(PrimePi[k] - PrimePi[k - 1])*(PrimePi[j] - PrimePi[j - 1]) (PrimePi[i] - PrimePi[i - 1]) (PrimePi[n - i - j - k] - PrimePi[n - i - j - k - 1]), {i, j, Floor[(n - j - k)/2]}], {j, k, Floor[(n - k)/3]}], {k, Floor[n/4]}], {n, 0, 50}]
Table[Total[Flatten[Select[IntegerPartitions[n,{4}],AllTrue[#,PrimeQ]&]]],{n,0,60}] (* Harvey P. Dale, Sep 28 2024 *)

Formula

a(n) = n * Sum_{k=1..floor(n/4)} Sum_{j=k..floor((n-k)/3)} Sum_{i=j..floor((n-j-k)/2)} c(i) * c(j) * c(k) * c(n-i-j-k), where c = A010051.
a(n) = n * A259194(n).
a(n) = A308771(n) + A308772(n) + A308773(n) + A308774(n).
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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