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.

A241743 Number of partitions p of n such that (number of numbers in p of form 3k) < (number of numbers in p of form 3k+1).

Original entry on oeis.org

0, 1, 1, 2, 3, 4, 6, 8, 12, 16, 21, 30, 40, 52, 72, 91, 121, 159, 202, 260, 335, 421, 535, 674, 840, 1052, 1304, 1614, 1996, 2451, 3002, 3674, 4468, 5442, 6592, 7971, 9624, 11584, 13898, 16691, 19947, 23823, 28410, 33782, 40113, 47610, 56302, 66572, 78569
Offset: 0

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Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 12 partitions: 71, 521, 5111, 44, 431, 422, 4211, 41111, 22211, 221111, 2111111, 11111111.

Links

Crossrefs

Cf. A241737, A241740, A241744, A241745.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[0, p] < s[2, p]], {n, 0, z}]  (* A241743 *)
Table[Count[f[n], p_ /; s[0, p] == s[1, p]], {n, 0, z}] (* A241744 *)
Table[Count[f[n], p_ /; s[0, p] > s[1, p]], {n, 0, z}]  (* A241745 *)

Formula

a(n) + A241744(n) + A241845(n) = A000041(n) for n >= 0.

A241744 Number of partitions p of n such that (number of numbers in p of form 3k) = (number of numbers in p of form 3k+1).

Original entry on oeis.org

1, 0, 1, 0, 2, 2, 3, 6, 7, 10, 17, 18, 27, 36, 44, 61, 76, 93, 124, 151, 193, 241, 297, 369, 462, 558, 707, 850, 1044, 1281, 1561, 1884, 2323, 2761, 3367, 4050, 4857, 5826, 7024, 8307, 9982, 11840, 14058, 16684, 19785, 23265, 27585, 32379, 38125, 44760
Offset: 1

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 7 partitions:  8, 611, 3311, 3221, 32111, 311111, 2222.

Crossrefs

Cf. A241737, A241740, A241743, A241745.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[0, p] < s[2, p]], {n, 0, z}]  (* A241743 *)
Table[Count[f[n], p_ /; s[0, p] == s[1, p]], {n, 0, z}] (* A241744 *)
Table[Count[f[n], p_ /; s[0, p] > s[1, p]], {n, 0, z}]  (* A241745 *)

Formula

a(n) + A241744(n) + A241845(n) = A000041(n) for n >= 0.

A241745 Number of partitions p of n such that (number of numbers in p of form 3k) > (number of numbers in p of form 3k+1).

Original entry on oeis.org

0, 0, 0, 1, 0, 1, 2, 1, 3, 4, 4, 8, 10, 13, 19, 24, 34, 45, 59, 79, 99, 130, 170, 212, 273, 348, 425, 546, 678, 833, 1041, 1284, 1558, 1940, 2351, 2862, 3496, 4227, 5093, 6187, 7409, 8920, 10706, 12795, 15277, 18259, 21671, 25803, 30579, 36218, 42836, 50596
Offset: 0

Views

Author

Clark Kimberling, Apr 28 2014

Keywords

Comments

Each number in p is counted once, regardless of its multiplicity.

Examples

			a(8) counts these 3 partitions:  62, 53, 332.

Crossrefs

Cf. A241737, A241740, A241743, A241744.

Programs

  • Mathematica
    z = 40; f[n_] := f[n] = IntegerPartitions[n]; s[k_, p_] := Count[Mod[DeleteDuplicates[p], 3], k];
Table[Count[f[n], p_ /; s[0, p] < s[2, p]], {n, 0, z}]  (* A241743 *)
Table[Count[f[n], p_ /; s[0, p] == s[1, p]], {n, 0, z}] (* A241744 *)
Table[Count[f[n], p_ /; s[0, p] > s[1, p]], {n, 0, z}]  (* A241745 *)

Formula

a(n) + A241744(n) + A241845(n) = A000041(n) for n >= 0.

A280419 Primes in A132934.

Original entry on oeis.org

1468910121415161820212224252627283032333435363839, 14689101214151618202122242526272830323334353638394042444546484950515254555657586062636465666869
Offset: 1

Views

Author

Sergey Pavlov, Jan 02 2017

Keywords

Comments

Primes that can be obtained from concatenation of the nonprime numbers.
In other words: primes that can be obtained from concatenation of 1 and the next consecutive composite numbers.
For a(n), the smallest prime divisor is a(n) and a(n) is a term of A241845.

Links

  • Prime Curios!, 39

Crossrefs

Cf. A002808, A018252, A132934, A241845.
Showing 1-4 of 4 results.

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