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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A305103 Heinz numbers of connected integer partitions with z-density -1.

Original entry on oeis.org

2, 3, 5, 7, 9, 11, 13, 17, 19, 21, 23, 25, 27, 29, 31, 37, 39, 41, 43, 47, 49, 53, 57, 59, 61, 63, 65, 67, 71, 73, 79, 81, 83, 87, 89, 91, 97, 101, 103, 107, 109, 111, 113, 115, 117, 121, 125, 127, 129, 131, 133, 137, 139, 147, 149, 151, 157, 159, 163, 167, 171
Offset: 1

Views

Author

Gus Wiseman, May 25 2018

Keywords

Comments

First differs from A305078 at a(61) = 171, A305078(61) = 169.
The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph.
The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)) where omega = A001221 is number of distinct prime factors.

Examples

			195 is the Heinz number of {2,3,6} with corresponding multiset multisystem {{1},{2},{1,2}}, which is connected with z-density -1.

Crossrefs

Cf. A001221, A056239, A112798, A286518, A302242, A303837, A304714, A304716, A305052, A305078, A305079.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zens[n_]:=If[n==1,0,Total@Cases[FactorInteger[n],{p_,k_}:>k*(PrimeNu[PrimePi[p]]-1)]-PrimeNu[LCM@@Cases[FactorInteger[n],{p_,k_}:>PrimePi[p]]]];
Select[Range[300],And[zens[#]==-1,Length[zsm[primeMS[#]]]==1]&]

A305504 Heinz numbers of integer partitions whose distinct parts plus 1 are connected.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 22, 23, 25, 27, 29, 31, 32, 33, 34, 37, 40, 41, 43, 44, 46, 47, 49, 50, 53, 55, 57, 59, 61, 62, 64, 66, 67, 68, 71, 73, 79, 80, 81, 82, 83, 85, 88, 89, 92, 93, 94, 97, 99, 100, 101, 103, 107, 109, 110, 113, 115
Offset: 1

Views

Author

Gus Wiseman, Jun 03 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).
Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A partition y is said to be connected if G(U(y + 1)) is a connected graph, where U(y + 1) is the set of distinct successors of the parts of y.
This is intended to be a cleaner form of A305078, where the treatment of empty multisets is arbitrary.

Examples

			The sequence of entries together with the corresponding twice-prime-factored multiset partitions (see A275024) begins:
   1: {}
   2: {{1}}
   3: {{2}}
   4: {{1},{1}}
   5: {{1,1}}
   7: {{3}}
   8: {{1},{1},{1}}
   9: {{2},{2}}
  10: {{1},{1,1}}
  11: {{1,2}}
  13: {{4}}
  16: {{1},{1},{1},{1}}
  17: {{1,1,1}}
  19: {{2,2}}
  20: {{1},{1},{1,1}}
  22: {{1},{1,2}}

Crossrefs

Cf. A001221, A048143, A056239, A112798, A275024, A286518, A290103, A302242, A304714, A304716, A305052, A305078, A305079, A305501.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[300],Length[zsm[primeMS[#]+1]]<=1&]

A305761 Nonprime Heinz numbers of z-trees.

Original entry on oeis.org

91, 203, 247, 299, 301, 377, 427, 551, 553, 559, 611, 689, 703, 707, 791, 817, 851, 923, 949, 973, 1027, 1073, 1081, 1141, 1159, 1247, 1267, 1313, 1339, 1349, 1363, 1391, 1393, 1501, 1537, 1591, 1603, 1679, 1703, 1739, 1757, 1769, 1781, 1807, 1897, 1919, 1961
Offset: 1

Views

Author

Gus Wiseman, Jun 10 2018

Keywords

Comments

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). Given a finite set S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor. For example, G({6,14,15,35}) is a 4-cycle. A multiset S is said to be connected if G(S) is a connected graph. The z-density of a multiset S of positive integers is Sum_{s in S} (omega(s) - 1) - omega(lcm(S)). Finally, a z-tree of weight n is a connected strict integer partition of n with at least two pairwise indivisible parts and z-density -1.

Examples

			2639 is the Heinz number of {4,6,10}, a z-tree corresponding to the multiset system {{1,1},{1,2},{1,3}}.

Crossrefs

Cf. A030019, A056239, A112798, A286520, A302242, A303362, A303837, A304118, A304714, A304716, A305052, A305078, A305079, A305081.

Programs

  • Mathematica
    primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Select[Range[3000],With[{p=primeMS[#]},And[UnsameQ@@p,Length[p]>1,zensity[p]==-1,Length[zsm[p]]==1,Select[Tuples[p,2],UnsameQ@@#&&Divisible@@#&]=={}]]&]
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