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.

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A324752 Number of strict integer partitions of n not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 2, 3, 1, 4, 4, 4, 5, 6, 7, 10, 9, 12, 12, 16, 17, 22, 22, 26, 31, 35, 37, 46, 50, 55, 66, 70, 82, 90, 101, 114, 127, 143, 159, 172, 202, 215, 246, 267, 301, 327, 366, 402, 447, 491, 545, 600, 655, 722, 795, 875, 964, 1050, 1152, 1259, 1383
Offset: 0

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Author

Gus Wiseman, Mar 16 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.

Examples

			The a(2) = 1 through a(17) = 12 strict integer partitions (A...H = 10...17):
  2  3  4  5  6   7   8  9   A   B    C   D    E    F    G    H
              42  43     54  64  65   75  76   86   87   97   98
                  52     63  73  83   84  85   95   96   A6   A7
                         72  82  542  93  94   A4   A5   C4   B6
                                      A2  B2   B3   B4   D3   C5
                                          643  752  C3   E2   D4
                                               842  D2   763  E3
                                                    654  943  854
                                                    843  A42  863
                                                    852       872
                                                              A52
                                                              B42
An example for n = 60 is (19,14,13,7,5,2), with prime indices:
  19: {8}
  14: {1,4}
  13: {6}
   7: {4}
   5: {3}
   2: {1}
None of these prime indices {1,3,4,6,8} belong to the partition, as required.

Crossrefs

The subset version is A324742, with maximal case is A324763. The non-strict version is A324757. The Heinz number version is A324761. An infinite version is A304360.
Cf. A000720, A001462, A007097, A074971, A078374, A112798, A276625, A290822, A305713, A306844, A324764.
Cf. A324695, A324743, A324748, A324751, A324756, A324758.

Programs

  • Mathematica
    Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!MemberQ[#,1]&&Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]],{n,0,30}]

A324761 Heinz numbers of integer partitions not containing 1 or any prime indices of the parts.

Original entry on oeis.org

1, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 33, 35, 37, 41, 43, 47, 49, 51, 53, 57, 59, 61, 63, 65, 67, 71, 73, 77, 79, 81, 83, 85, 87, 89, 91, 93, 95, 97, 99, 101, 103, 107, 109, 113, 115, 121, 123, 125, 127, 129, 131, 133, 137, 139, 143, 147, 149
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

Examples

			The sequence of terms together with their prime indices begins:
   1: {}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  17: {7}
  19: {8}
  21: {2,4}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  35: {3,4}
  37: {12}
  41: {13}
  43: {14}

Crossrefs

The subset version is A324742, with maximal case A324763. The strict integer partition version is A324752. The integer partition version is A324757. An infinite version is A324695.
Cf. A000720, A001221, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324743, A324751, A324756, A324758, A324764.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1,100,2],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

A324841 Matula-Goebel numbers of fully recursively anti-transitive rooted trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 21, 23, 25, 27, 31, 32, 35, 49, 51, 53, 57, 59, 63, 64, 67, 73, 77, 81, 83, 85, 95, 97, 103, 115, 121, 125, 127, 128, 131, 133, 147, 149, 153, 159, 161, 171, 175, 177, 187, 189, 201, 209, 217, 227, 233, 241, 243, 245
Offset: 1

Views

Author

Gus Wiseman, Mar 17 2019

Keywords

Comments

An unlabeled rooted tree is fully recursively anti-transitive if no proper terminal subtree of any terminal subtree is a branch of the larger subtree.

Examples

			The sequence of fully recursively anti-transitive rooted trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  11: ((((o))))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  21: ((o)(oo))
  23: (((o)(o)))
  25: (((o))((o)))
  27: ((o)(o)(o))
  31: (((((o)))))
  32: (ooooo)
  35: (((o))(oo))
  49: ((oo)(oo))
  51: ((o)((oo)))
  53: ((oooo))
  57: ((o)(ooo))
  59: ((((oo))))
  63: ((o)(o)(oo))

Crossrefs

Cf. A000081, A007097, A290689, A303431, A304360, A306844, A316502, A318185, A318186.
Cf. A324695, A324751, A324756, A324758, A324766, A324768, A324769.
Cf. A324838, A324840, A324844.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
fratQ[n_]:=And[Intersection[Union@@Rest[FixedPointList[Union@@primeMS/@#&,primeMS[n]]],primeMS[n]]=={},And@@fratQ/@primeMS[n]];
Select[Range[100],fratQ]
Previous Showing 21-23 of 23 results.

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