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 11-18 of 18 results.

A347578 Number of partitions of n into at most 4 prime parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7, 8, 8, 9, 10, 9, 11, 11, 13, 11, 15, 12, 16, 15, 16, 15, 18, 16, 20, 17, 23, 18, 24, 20, 26, 22, 26, 23, 31, 23, 33, 26, 35, 26, 39, 27, 41, 32, 41, 31, 46, 31, 48, 34, 51, 34, 54, 36, 58, 40, 58, 42, 64, 41
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 08 2021

Keywords

Crossrefs

Cf. A000040, A000607, A071335, A117278, A259194, A347552, A347609, A347610.

Formula

a(n) = Sum_{k=1..4} A117278(n,k) for n >= 2. - Alois P. Heinz, Sep 08 2021

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).

A308774 Sum of the 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, 3, 3, 8, 8, 12, 17, 12, 19, 23, 30, 38, 54, 31, 62, 59, 79, 73, 119, 71, 151, 113, 169, 115, 207, 102, 234, 171, 263, 168, 350, 191, 425, 220, 391, 265, 518, 246, 606, 322, 636, 383, 774, 348, 918, 477, 947, 516, 1102, 468, 1259
Offset: 0

Views

Author

Wesley Ivan Hurt, Jun 23 2019

Keywords

Links

Crossrefs

Cf. A010051, A259194, A308771, A308772, A308773, A308809.

Programs

  • Mathematica
    Table[Sum[Sum[Sum[(n - i - j - 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) * (n-i-j-k), where c = A010051.
a(n) = A308809(n) - A308771(n) - A308772(n) - A308773(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).

A070757 Number of ways to write 4n as a sum of 4 primes.

Original entry on oeis.org

0, 1, 2, 3, 6, 7, 11, 13, 15, 21, 24, 26, 34, 38, 41, 50, 56, 57, 70, 77, 81, 96, 100, 109, 126, 129, 138, 156, 163, 175, 199, 198, 211, 239, 241, 258, 283, 281, 308, 335, 335, 360, 382, 385, 421, 445, 437, 482, 508, 503, 556, 571, 561, 632, 657, 645, 705, 726
Offset: 1

Views

Author

Benoit Cloitre, May 14 2002

Keywords

Examples

			4*3 = 12 = 2+2+3+5 = 3+3+3+3 hence a(3)=2.

Links

Crossrefs

Cf. A259194.

Programs

  • Maple
    N:= 400: # to get a(0) to a(N/4)
Primes:= select(isprime, [$1..N]):
np:= nops(Primes):
for j from 0 to np do g[0, j]:= 1 od:
for n from 1 to 4 do
  g[n, 0]:= 0:
  for j from 1 to np do
     g[n, j]:= convert(series(add(g[k, j-1]
          *x^((n-k)*Primes[j]), k=0..n), x, N+1), polynom)
  od
od:
seq(coeff(g[4, np], x, 4*i), i=1..N/4); # Robert Israel, Oct 29 2019
  • PARI
    a(n)={my(s=0); forprime(i=2, n, forprime(j=i, (4*n-i)\3, forprime(k=j, (4*n-i-j)\2, if(isprime(4*n-i-j-k), s++)))); s} \\ Andrew Howroyd, Oct 29 2019

Formula

a(n) = A259194(4*n). - Robert Israel, Oct 29 2019

Extensions

Corrected by Robert Israel, Oct 29 2019

A350563 Sums of 4 primes whose product is the average of a twin prime pair.

Original entry on oeis.org

12, 15, 19, 23, 26, 29, 36, 37, 43, 51, 53, 57, 61, 67, 69, 79, 81, 83, 89, 99, 107, 109, 113, 119, 121, 123, 129, 131, 133, 137, 141, 143, 146, 149, 153, 156, 159, 167, 169, 173, 177, 187, 189, 197, 201, 207, 209, 213, 223, 227, 229, 233, 237, 239, 247, 249, 251, 259, 261, 263
Offset: 1

Views

Author

Wesley Ivan Hurt, Jan 05 2022

Keywords

Examples

			12 is in the sequence since 12 = 2+2+3+5 (all primes) and 2*2*3*5 = 60 is the average of the twin prime pair (59,61).

Links

Crossrefs

Cf. A001097, A014574, A259194.
Previous Showing 11-18 of 18 results.

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