A340020 MM-numbers of labeled graphs with loops, without isolated vertices.

1, 7, 13, 23, 29, 43, 47, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 553, 559, 577, 607, 611, 631, 647, 653, 661, 667, 673, 677

Offset

1,2

Comments

Here a loop is an edge with two equal vertices, distinguished from a half-loop, which has only one vertex.

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 of multisets 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 of multisets with MM-number 78 is {{},{1},{1,2}}.

Also products of distinct primes whose prime indices are semiprimes, where a semiprime (A001358) is a product of any two prime numbers.

Links

Table of n, a(n) for n=1..55.

Example

The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
      1: {}              161: {{1,1},{2,2}}    347: {{2,9}}
      7: {{1,1}}         163: {{1,8}}          373: {{1,12}}
     13: {{1,2}}         167: {{2,6}}          377: {{1,2},{1,3}}
     23: {{2,2}}         199: {{1,9}}          389: {{4,5}}
     29: {{1,3}}         203: {{1,1},{1,3}}    421: {{1,13}}
     43: {{1,4}}         227: {{4,4}}          439: {{3,7}}
     47: {{2,3}}         233: {{2,7}}          443: {{1,14}}
     73: {{2,4}}         257: {{3,5}}          449: {{2,10}}
     79: {{1,5}}         269: {{2,8}}          467: {{4,6}}
     91: {{1,1},{1,2}}   271: {{1,10}}         487: {{2,11}}
     97: {{3,3}}         293: {{1,11}}         491: {{1,15}}
    101: {{1,6}}         299: {{1,2},{2,2}}    499: {{3,8}}
    137: {{2,5}}         301: {{1,1},{1,4}}    511: {{1,1},{2,4}}
    139: {{1,7}}         313: {{3,6}}          553: {{1,1},{1,5}}
    149: {{3,4}}         329: {{1,1},{2,3}}    559: {{1,2},{1,4}}

Mathematica

Select[Range[100], SquareFreeQ[#]&&FreeQ[If[#==1, {}, FactorInteger[#]], {p_, k_}/; PrimeOmega[PrimePi[p]]!=2]&]

Crossrefs

The case with only one edge is A106349.

The case covering an initial interval is A320461.

The version allowing multiple edges is A339112.

The half-loop version covering an initial interval is A340018.

The half-loop version is A340019.

A006450 lists primes of prime index.

A302242 is the weight of the multiset of multisets with MM-number n.

A302494 lists MM-numbers of sets of sets, with connected case A328514.

A309356 lists MM-numbers of simple graphs.

A339113 lists MM-numbers of multigraphs.

Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326754, A326788.

Sequence in context: A214794 A043104 A339112 * A106349 A293657 A048449

Adjacent sequences: A340017 A340018 A340019 * A340021 A340022 A340023

Keyword

nonn

Author

Gus Wiseman, Jan 02 2021

Status

approved