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 1-5 of 5 results.

A241409 Number of partitions of n such that the number of parts having multiplicity >1 is a part and the number of distinct parts is a part.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 3, 5, 6, 9, 12, 16, 23, 26, 39, 45, 67, 78, 106, 130, 171, 207, 270, 329, 419, 516, 637, 787, 978, 1190, 1451, 1775, 2166, 2613, 3173, 3827, 4613, 5537, 6659, 7948, 9523, 11316, 13505, 16014, 19059, 22455, 26667, 31376, 37079, 43501, 51282
Offset: 0

Views

Author

Clark Kimberling, Apr 22 2014

Keywords

Comments

As used here, the term "distinct parts" includes each number, once, that occurs more than once; e.g., the distinct parts of the partition {4,3,3,1,1,1} are 4, 3, 1.

Examples

			a(6) counts these 3 partitions:  411, 3111, 21111.

Crossrefs

Cf. A241408, A241410, A241411, A241412.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)

A241408 a(n) is the number of partitions of n such that the number of parts having multiplicity > 1 is a part.

Original entry on oeis.org

0, 0, 1, 1, 2, 4, 5, 9, 11, 18, 24, 34, 46, 63, 83, 109, 147, 189, 245, 315, 406, 513, 650, 817, 1030, 1287, 1593, 1978, 2450, 3013, 3689, 4523, 5511, 6711, 8140, 9852, 11892, 14334, 17217, 20657, 24727, 29531, 35197, 41894, 49761, 59000, 69861, 82542, 97393
Offset: 0

Views

Author

Clark Kimberling, Apr 22 2014

Keywords

Examples

			a(6) counts these 5 partitions: 411, 3111, 2211, 21111, 111111; e.g., the number of parts of 2211 that have multiplicity > 1 is 2, which counts 1 (with multiplicity 2) and 2 (also with multiplicity 2), so that 2211 is a term because 2 is a part.

Crossrefs

Cf. A241409, A241410, A241411, A241412.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)

A241410 Number of partitions of n such that the number of parts having multiplicity >1 is not a part and the number of distinct parts is a part.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 2, 2, 4, 5, 7, 8, 12, 17, 22, 29, 33, 49, 59, 77, 97, 123, 153, 199, 234, 306, 375, 460, 557, 708, 845, 1048, 1266, 1548, 1852, 2282, 2698, 3303, 3919, 4732, 5634, 6786, 7991, 9598, 11343, 13502, 15897, 18912, 22180, 26298, 30775, 36259
Offset: 0

Views

Author

Clark Kimberling, Apr 22 2014

Keywords

Comments

As used here, the term "distinct parts" includes each number, once, that occurs more than once; e.g., the distinct parts of the partition {4,3,3,1,1,1} are 4, 3, 1.

Examples

			a(6) counts these 2 partitions: 42, 321.

Crossrefs

Cf. A241408, A241409, A241411, A241412.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)

A241411 Number of partitions of n such that the number of parts having multiplicity >1 is a part and the number of distinct parts is not a part.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 2, 4, 5, 9, 12, 18, 23, 37, 44, 64, 80, 111, 139, 185, 235, 306, 380, 488, 611, 771, 956, 1191, 1472, 1823, 2238, 2748, 3345, 4098, 4967, 6025, 7279, 8797, 10558, 12709, 15204, 18215, 21692, 25880, 30702, 36545, 43194, 51166, 60314, 71255
Offset: 0

Views

Author

Clark Kimberling, Apr 22 2014

Keywords

Comments

As used here, the term "distinct parts" includes each number, once, that occurs more than once; e.g., the distinct parts of the partition {4,3,3,1,1,1} are 4, 3, 1.

Examples

			a(6) counts these 2 partitions:  411, 3111.

Crossrefs

Cf. A241408, A241409, A241410, A241412.

Programs

  • Mathematica
    z = 30; f[n_] := f[n] = IntegerPartitions[n]; e[p_] := Length[DeleteDuplicates[Select[p, Count[p, #] > 1 &]]]; d[p_] := Length[DeleteDuplicates[p]];
Table[Count[f[n], p_ /; MemberQ[p, e[p]]], {n, 0, z}]  (* A241408 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && MemberQ[p, d[p]]], {n, 0, z}]  (* A241409 *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && MemberQ[p, d[p]] ], {n, 0, z}] (* A241410 *)
Table[Count[f[n], p_ /; MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241411  *)
Table[Count[f[n], p_ /; ! MemberQ[p, e[p]] && ! MemberQ[p, d[p]] ], {n, 0, z}] (* A241412  *)

A251556 Conjectured values of n which are the minimal cycle representatives of finite cycles in the permutation of the natural numbers defined by A098550.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 12, 17, 24, 40, 42, 50, 86, 107
Offset: 1

Views

Author

N. J. A. Sloane, Dec 23 2014

Keywords

Comments

At present we do not know for certain that 11, 29, 36, etc. are really missing, since the status of the cycle containing 11 is not known. It appears to be infinite (see A241412).
For other values that are known to be in the sequence see the Havermann link.
A251411 is a subsequence.

Links

Crossrefs

Cf. A098550, A251411, A251412.
Showing 1-5 of 5 results.

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

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