proposed

approved

editing

proposed

MM-numbers of simple labeled simple covering graphs.

The covering case is A320458.

Cf. A001222, A001358, A006129, A056239, A112798, A302242, A320458, A322551.

Covering means there are no isolated vertices, i.e., the vertex set is the union of the edge set.

The sequence of edge- sets together with their MM-numbers begins:

MM-numbers of simple labeled covering graphs.

Covering means there are no isolated vertices.

The sequence of graphs edge-sets together with their MM-numbers begins:

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

allocated for Gus WisemanMM-numbers of simple labeled graphs.

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 559, 577, 607, 611, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 949

1,2

First differs from A322551 in having 377.

A multiset multisystem is a finite multiset of finite multisets. 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 multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

Also products of distinct elements of A322551.

The sequence of graphs together with their MM-numbers begins:

1: {}

13: {{1,2}}

29: {{1,3}}

43: {{1,4}}

47: {{2,3}}

73: {{2,4}}

79: {{1,5}}

101: {{1,6}}

137: {{2,5}}

139: {{1,7}}

149: {{3,4}}

163: {{1,8}}

167: {{2,6}}

199: {{1,9}}

233: {{2,7}}

257: {{3,5}}

269: {{2,8}}

271: {{1,10}}

293: {{1,11}}

313: {{3,6}}

347: {{2,9}}

373: {{1,12}}

377: {{1,2},{1,3}}

389: {{4,5}}

421: {{1,13}}

primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Select[Range[1000], And[SquareFreeQ[#], And@@(And[SquareFreeQ[#], Length[primeMS[#]]==2]&/@primeMS[#])]&]

Simple graphs are A006125.

The covering case is A320458.

The case for BII-numbers is A326788.

Cf. A001222, A001358, A006129, A056239, A112798, A302242, A322551.

allocated

nonn

Gus Wiseman, Jul 25 2019

approved

editing

allocated for Gus Wiseman

recycled

allocated