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A309356 MM-numbers of labeled simple covering graphs. 14
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

First differs from A322551 in having 377.

Also products of distinct elements of A322551.

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}}.

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

LINKS

Table of n, a(n) for n=1..54.

EXAMPLE

The sequence of edge sets 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}}

MATHEMATICA

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[#])]&]

CROSSREFS

Simple graphs are A006125.

The case for BII-numbers is A326788.

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

Sequence in context: A120827 A320631 A339113 * A322551 A228069 A044074

Adjacent sequences:  A309353 A309354 A309355 * A309357 A309358 A309359

KEYWORD

nonn

AUTHOR

Gus Wiseman, Jul 25 2019

STATUS

approved

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Last modified September 20 03:47 EDT 2021. Contains 347577 sequences. (Running on oeis4.)