A320458 -id:A320458 - cp's OEIS frontend

A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

1, 2, 3, 4, 6, 7, 8, 9, 12, 13, 14, 15, 16, 18, 19, 21, 24, 26, 27, 28, 30, 32, 35, 36, 37, 38, 39, 42, 45, 48, 49, 52, 53, 54, 56, 57, 60, 61, 63, 64, 65, 69, 70, 72, 74, 75, 76, 78, 81, 84, 89, 90, 91, 95, 96, 98, 104, 105, 106, 108, 111, 112, 113, 114, 117

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

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 n-th multiset multisystem 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 78th multiset multisystem is {{},{1},{1,2}}.

			The sequence of terms together with their multiset multisystems begins:
   1: {}
   2: {{}}
   3: {{1}}
   4: {{},{}}
   6: {{},{1}}
   7: {{1,1}}
   8: {{},{},{}}
   9: {{1},{1}}
  12: {{},{},{1}}
  13: {{1,2}}
  14: {{},{1,1}}
  15: {{1},{2}}
  16: {{},{},{},{}}
  18: {{},{1},{1}}
  19: {{1,1,1}}
  21: {{1},{1,1}}
  24: {{},{},{},{1}}
  26: {{},{1,2}}
  27: {{1},{1},{1}}
  28: {{},{},{1,1}}
  30: {{},{1},{2}}
  32: {{},{},{},{},{}}

Cf. A001222, A003963, A034691, A034729, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320458, A320459, A320461, A320462, A320463, A320464, A320532, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[100],normQ[primeMS/@primeMS[#]]&]

A320922 Heinz numbers of graphical partitions.

Original entry on oeis.org

1, 4, 12, 16, 27, 36, 40, 48, 64, 81, 90, 108, 112, 120, 144, 160, 192, 225, 243, 252, 256, 270, 300, 324, 336, 352, 360, 400, 432, 448, 480, 567, 576, 625, 630, 640, 675, 729, 750, 756, 768, 792, 810, 832, 840, 900, 972, 1000, 1008, 1024, 1056, 1080, 1120

Offset: 1

Gus Wiseman, Oct 24 2018

nonn

The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).

An integer partition is graphical if it comprises the vertex-degrees of some simple graph.

			The sequence of all graphical partitions begins: (), (11), (211), (1111), (222), (2211), (3111), (21111), (111111), (2222), (3221), (22211), (41111), (32111), (221111), (311111), (2111111), (3322), (22222), (42211).

Cf. A000070, A000569, A007717, A056239, A096373, A112798, A147878, A209816, A300061, A320458, A320911, A320923, A320924.

prptns[m_]:=Union[Sort/@If[Length[m]==0,{{}},Join@@Table[Prepend[#,m[[ipr]]]&/@prptns[Delete[m,List/@ipr]],{ipr,Select[Prepend[{#},1]&/@Select[Range[2,Length[m]],m[[#]]>m[[#-1]]&],UnsameQ@@m[[#]]&]}]]];
Select[Range[1000],Select[prptns[Flatten[MapIndexed[Table[#2,{#1}]&,If[#==1,{},Flatten[Cases[FactorInteger[#],{p_,k_}:>Table[PrimePi[p],{k}]]]]]]],UnsameQ@@#&]!={}&]

A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

Original entry on oeis.org

1, 1, 2, 4, 11, 31, 102, 342, 1213, 4361, 16016, 59348, 222117, 836315, 3166852, 12042620, 45967479, 176005709, 675759564, 2600672458, 10029832754, 38753710486, 149990133774, 581393603996, 2256710139346, 8770547818956, 34125389919850, 132919443189544, 518232001761434, 2022337118015338, 7898574056034636, 30873421455729728

Offset: 0

N. J. A. Sloane

In other words, a(n) is the number of graphic sequences of length n, where a graphic sequence is a sequence of numbers which can be the degree sequence of some graph.

In the article by A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, "Parallel enumeration of degree sequences of simple graphs II", in Table 4 on page 260 the values a(30) = 7898574056034638 and a(31) = 30873429530206738 are incorrect due to the incorrect Gz(30) = 5876236938019300 and Gz(31) = 22974847474172100. - Wang Kai, Jun 05 2016

			For n = 3, there are 4 different graphic sequences possible: 0 0 0; 1 1 0; 2 1 1; 2 2 2. - Daan van Berkel (daan.v.berkel.1980(AT)gmail.com), Jun 25 2010
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(4) = 11 sorted degree sequences:
  ()  (0)  (0,0)  (0,0,0)  (0,0,0,0)
           (1,1)  (0,1,1)  (0,0,1,1)
                  (1,1,2)  (0,1,1,2)
                  (2,2,2)  (0,2,2,2)
                           (1,1,1,1)
                           (1,1,1,3)
                           (1,1,2,2)
                           (1,2,2,3)
                           (2,2,2,2)
                           (2,2,3,3)
                           (3,3,3,3)
For example, the graph {{2,3},{2,4}} has degrees (0,2,1,1), so (0,1,1,2) is counted under a(4).
(End)

R. A. Brualdi and H. J. Ryser, Combinatorial Matrix Theory, Cambridge Univ. Press, 1992.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978).

Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Table of n, a(n) for n = 0..1651 (Terms 1 through 31 were computed by various authors; terms 32 through 34 by Axel Kohnert and Wang Kai; terms 35 to 79 by Wang Kai)
Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Counting graphic sequences via integrated random walks, arXiv:2301.07022 [math.CO], 2023.
T. M. Barnes and C. D. Savage, A recurrence for counting graphical partitions, Electronic J. Combinatorics, 2 (1995), #R11.
B. A. Chat, S. Pirzada, and A. Iványi, Recognition of split-graphic sequences, Acta Universitatis Sapientiae, Informatica, 6, 2 (2014) 252-286.
D. Dimitrov, Efficient computation of trees with minimal atom-bond connectivity index, arXiv:1305.1155 [cs.DM], 2013.
A. Iványi, L. Lucz, T. F. Móri and P. Sótér, On Erdős-Gallai and Havel-Hakimi algorithms, Acta Univ. Sapientiae, Inform. 3(2) (2011), 230-268.
A. Ivanyi, L. Lucz, T. Matuszka, and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapientiae, Informatica, 4, 2 (2012), 260-288. - From _N. J. A. Sloane_, Feb 15 2013
A. Ivanyi and J. E. Schoenfield, Deciding football sequences, Acta Univ. Sapientiae, Informatica, 4, 1 (2012), 130-183. - From _N. J. A. Sloane_, Dec 22 2012 [Disclaimer: I am not one of the authors of this paper; I was unpleasantly surprised to find my name on it, as explained here. - _Jon E. Schoenfield_, Nov 26 2016]
Wang Kai, Efficient Counting of Degree Sequences, arXiv:1604.04148 [math.CO], 2016. Gives 79 terms.
P. W. Mills, R. P. Rundle, V. M. Dwyer, T. Tilma, and S. J. Devitt, A proposal for an efficient quantum algorithm solving the graph isomorphism problem, arXiv 1711.09842, 2017.
P. R. Stein, On the number of graphical partitions, pp. 671-684 of Proc. 9th S-E Conf. Combinatorics, Graph Theory, Computing, Congr. Numer. 21 (1978). [Annotated scanned copy]
Eric Weisstein's World of Mathematics, Degree Sequence.
Eric Weisstein's World of Mathematics, Graphic Sequence.
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.
Index entries for sequences related to graphical partitions

Counting the positive partitions by sum gives A000569, ranked by A320922.
The version with half-loops is A029889, with covering case A339843.
The covering case (no zeros) is A095268.
Covering simple graphs are ranked by A309356 and A320458.
Non-graphical partitions are counted by A339617 and ranked by A339618.
The version with loops is A339844, with covering case A339845.
A006125 counts simple graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A320921 counts connected graphical partitions.
A322353 counts factorizations into distinct semiprimes.
A339659 counts graphical partitions of 2n into k parts.
A339661 counts factorizations into distinct squarefree semiprimes.
Cf. A004250, A007717, A181819, A320894, A320923, A339560, A339561, A339842.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Subsets[Subsets[Range[n],{2}]]]],{n,0,5}] (* Gus Wiseman, Dec 31 2020 *)

G.f. = 1 + x + 2*x^2 + 4*x^3 + 11*x^4 + 31*x^5 + 102*x^6 + 342*x^7 + 1213*x^8 + ...

a(n) ~ c * 4^n / n^(3/4) for some constant c > 0. Computational estimates suggest c ≈ 0.099094. - Tom Johnston, Jan 18 2023

More terms from Torsten Sillke, torsten.sillke(AT)lhsystems.com, using Cor. 6.3.3, Th. 6.3.6, Cor. 6.2.5 of Brualdi-Ryser.
a(19) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 19 2007
a(20)-a(23) from Nathann Cohen, Jul 09 2011
a(24)-a(29) from Antal Iványi, Nov 15 2011
a(30) and a(31) corrected by Wang Kai, Jun 05 2016

A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 91, 161, 299, 329, 377, 611, 667, 1261, 1363, 1937, 2021, 2093, 2117, 2639, 4277, 4669, 7567, 8671, 8827, 9541, 13559, 14053, 14147, 14819, 15617, 16211, 17719, 23989, 24017, 26273, 27521, 28681, 29003, 31349, 31913, 36569, 44551, 44603, 46483, 48691

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

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}}.

			The sequence of terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   667: {{2,2},{1,3}}
  1261: {{3,3},{1,2}}
  1363: {{1,3},{2,3}}
  1937: {{1,2},{3,4}}
  2021: {{1,4},{2,3}}
  2093: {{1,1},{2,2},{1,2}}
  2117: {{1,3},{2,4}}
  2639: {{1,1},{1,2},{1,3}}
  4277: {{1,1},{1,2},{2,3}}
  4669: {{1,1},{2,2},{1,3}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A305052.

Cf. A320456, A320458, A320459, A320462, A320532.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

A309356 MM-numbers of labeled simple covering graphs.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jul 25 2019

nonn

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.

			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}}

Simple graphs are A006125.
The case for BII-numbers is A326788.
Cf. A001222, A001358, A006129, A056239, A112798, A302242, A320458, A322551.

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[#])]&]

A095268 Number of distinct degree sequences among all n-vertex graphs with no isolated vertices.

Original entry on oeis.org

1, 0, 1, 2, 7, 20, 71, 240, 871, 3148, 11655, 43332, 162769, 614198, 2330537, 8875768, 33924859, 130038230, 499753855, 1924912894, 7429160296, 28723877732, 111236423288, 431403470222, 1675316535350, 6513837679610, 25354842100894, 98794053269694, 385312558571890, 1504105116253904, 5876236938019298, 22974847399695092

Offset: 0

Eric W. Weisstein, May 31 2004

nonn

A002494 is the number of graphs on n nodes with no isolated points and A095268 is the number of these graphs having distinct degree sequences.
Now that more terms have been computed, we can see that this is not the self-convolution of any integer sequence. - Paul D. Hanna, Aug 18 2006
Is it true that a(n+1)/a(n) tends to 4? Is there a heuristic argument why this might be true? - Gordon F. Royle, Aug 29 2006
Previous values a(30) = 5876236938019300 from Lorand Lucz, Jul 07 2013 and a(31) = 22974847474172100 from Lorand Lucz, Sep 03 2013 are wrong. New values a(30) and a(31) independently computed Kai Wang and Axel Kohnert. - Vaclav Kotesovec, Apr 15 2016
In the article by A. Iványi, G. Gombos, L. Lucz, T. Matuszka: "Parallel enumeration of degree sequences of simple graphs II" is in the tables on pages 258 and 261 a wrong value a(31) = 22974847474172100, but in the abstract another wrong value a(31) = 22974847474172374. - Vaclav Kotesovec, Apr 15 2016
The asymptotic formula given below confirms that a(n+1)/a(n) tends to 4. - Tom Johnston, Jan 18 2023

			a(4) = 7 because a 4-vertex graph with no isolated vertices can have degree sequence 1111, 2211, 2222, 3111, 3221, 3322 or 3333.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(0) = 1 through a(3) = 7 sorted degree sequences (empty column indicated by dot):
  ()  .  (1,1)  (2,1,1)  (1,1,1,1)
                (2,2,2)  (2,2,1,1)
                         (2,2,2,2)
                         (3,1,1,1)
                         (3,2,2,1)
                         (3,3,2,2)
                         (3,3,3,3)
For example, the complete graph K_4 has degrees y = (3,3,3,3), so y is counted under a(4). On the other hand, the only half-loop-graphs (up to isomorphism) with degrees y = (4,2,2,1) are: {(1),(1,2),(1,3),(1,4),(2,3)} and {(1),(2),(3),(1,2),(1,3),(1,4)}; and since neither of these is a graph (due to having half-loops), y is not counted under a(4).
(End)

Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Table of n, a(n) for n = 0..1651 (terms 0 through 79 from Kai Wang)
Paul Balister, Serte Donderwinkel, Carla Groenland, Tom Johnston, and Alex Scott, Counting graphic sequences via integrated random walks, arXiv:2301.07022 [math.CO], 2023.
A. Iványi, L. Lucz, T. Matuszka and S. Pirzada, Parallel enumeration of degree sequences of simple graphs, Acta Univ. Sapiantiae, Inform.4 (2) (2012) 260-288.
A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, Parallel enumeration of degree sequences of simple graphs II, Acta Universitatis Sapientiae, Informatica, Volume 5, Issue 2 (Dec 2013).
A. Iványi, L. Lucz, T. F. Móri and P. Sótér, On Erdős-Gallai and Havel-Hakimi algorithms, Acta Univ. Sapiantiae, Inform. 3 (2) (2011) 230-268.
A. Kohnert, Number of different degree sequences of a graph with no isolated vertices, 2016.
Frank Ruskey, Alley CATs in Search of Good Homes
Kai Wang, Efficient Counting of Degree Sequences, arXiv:1604.04148 [math.CO], 2016. Gives 79 terms. But a(30) and a(31) are different.
Eric Weisstein's World of Mathematics, Degree sequence
Gus Wiseman, Counting and ranking factorizations, factorability, and vertex-degree partitions for groupings into pairs.

Cf. A002494, A004250, A007721 (analog for connected graphs), A271831.
Counting the same partitions by sum gives A000569.
Allowing isolated nodes gives A004251.
The version with half-loops is A029889, with covering case A339843.
Covering simple graphs are ranked by A309356 and A320458.
Graphical partitions are ranked by A320922.
The version with loops is A339844, with covering case A339845.
A006125 counts simple graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A058696 counts partitions of even numbers, ranked by A300061.
A339659 is a triangle counting graphical partitions.
Cf. A007717, A096373, A181819, A320921, A322661, A339559, A339560.

Table[Length[Union[Sort[Table[Count[Join@@#,i],{i,n}]]&/@Select[Subsets[Subsets[Range[n],{2}]],Union@@#==Range[n]&]]],{n,0,5}] (* Gus Wiseman, Dec 31 2020 *)

a(n) ~ c * 4^n / n^(3/4) for some c > 0. Computational estimates suggest c ≈ 0.074321. - Tom Johnston, Jan 18 2023

Edited by N. J. A. Sloane, Aug 26 2006
More terms from Gordon F. Royle, Aug 21 2006
a(21) and a(22) from Frank Ruskey, Aug 29 2006
a(23) from Frank Ruskey, Aug 31 2006
a(24)-a(29) from Matuszka Tamás, Jan 10 2013
a(30)-a(31) from articles by Kai Wang and Axel Kohnert, Apr 15 2016
a(0) = 1 and a(1) = 0 prepended by Gus Wiseman, Dec 31 2020

A322551 Primes indexed by squarefree semiprimes.

Original entry on oeis.org

13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 199, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947, 983, 1051, 1061, 1093

Offset: 1

Gus Wiseman, Dec 15 2018

nonn

A squarefree semiprime is a product of two distinct prime numbers.

			The sequence of edges whose MM-numbers belong to the sequence begins: {{1,2}}, {{1,3}}, {{1,4}}, {{2,3}}, {{2,4}}, {{1,5}}, {{1,6}}, {{2,5}}, {{1,7}}, {{3,4}}, {{1,8}}, {{2,6}}, {{1,9}}, {{2,7}}, {{3,5}}, {{2,8}}.

Cf. A001358, A003963, A006881, A056239, A085156, A106349, A112798, A302242, A302491, A320458, A320459, A320461.

Select[Range[100],PrimeOmega[#]==1&&PrimeOmega[PrimePi[#]]==2&&SquareFreeQ[PrimePi[#]]&]

isok(p) = isprime(p) && (ip=primepi(p)) && (omega(ip)==2) && (bigomega(ip) == 2); \\ Michel Marcus, Dec 16 2018

A320462 MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.

Original entry on oeis.org

1, 7, 13, 49, 91, 161, 169, 299, 329, 343, 377, 611, 637, 667, 1127, 1183, 1261, 1363, 1937, 2021, 2093, 2117, 2197, 2303, 2401, 2639, 3703, 3887, 4277, 4459, 4669, 4901, 6877, 7567, 7889, 7943, 8281, 8671, 8827, 9541, 10933, 13559, 14053, 14147, 14651, 14819

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

			The sequence of terms together with their multiset multisystems begins:
     1: {}
     7: {{1,1}}
    13: {{1,2}}
    49: {{1,1},{1,1}}
    91: {{1,1},{1,2}}
   161: {{1,1},{2,2}}
   169: {{1,2},{1,2}}
   299: {{2,2},{1,2}}
   329: {{1,1},{2,3}}
   343: {{1,1},{1,1},{1,1}}
   377: {{1,2},{1,3}}
   611: {{1,2},{2,3}}
   637: {{1,1},{1,1},{1,2}}
   667: {{2,2},{1,3}}
  1127: {{1,1},{1,1},{2,2}}
  1183: {{1,1},{1,2},{1,2}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A305052, A305078.

Cf. A320456, A320458, A320459, A320461, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[normQ[primeMS/@primeMS[#]],And@@(Length[primeMS[#]]==2&/@primeMS[#])]&]

A320459 MM-numbers of labeled multigraphs spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 169, 377, 611, 1363, 1937, 2021, 2117, 2197, 4901, 7943, 10933, 16211, 17719, 25181, 26273, 27521, 28561, 28717, 39527, 44603, 56173, 58609, 61393, 63713, 64061, 83291, 86903, 91031, 91039, 94987, 99499, 103259, 141401, 142129, 143663, 146653, 147533

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    169: {{1,2},{1,2}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1363: {{1,3},{2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
   2197: {{1,2},{1,2},{1,2}}
   4901: {{1,2},{1,2},{1,3}}
   7943: {{1,2},{1,2},{2,3}}
  10933: {{1,2},{1,3},{1,3}}
  16211: {{1,2},{1,3},{1,4}}
  17719: {{1,2},{1,3},{2,3}}
  25181: {{1,2},{1,2},{3,4}}
  26273: {{1,2},{1,4},{2,3}}
  27521: {{1,2},{1,3},{2,4}}
  28561: {{1,2},{1,2},{1,2},{1,2}}
  28717: {{1,2},{2,3},{2,3}}
  39527: {{1,3},{1,3},{2,3}}
  44603: {{1,2},{2,3},{2,4}}

Eric Weisstein's World of Mathematics, Simple Graph

Cf. A003963, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052, A305078.

Cf. A320456, A320458, A320461, A320462, A320464.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[100000],And[normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],Length[primeMS[#]]==2]&/@primeMS[#])]&]

A320463 MM-numbers of labeled simple hypergraphs with no singletons spanning an initial interval of positive integers.

Original entry on oeis.org

1, 13, 113, 377, 611, 1291, 1363, 1469, 1937, 2021, 2117, 3277, 4537, 4859, 5249, 5311, 7423, 8249, 8507, 16211, 16403, 16559, 16783, 16837, 17719, 20443, 20453, 24553, 25477, 26273, 26969, 27521, 34567, 37439, 39437, 41689, 42011, 42137, 42601, 43873, 43957

Offset: 1

Gus Wiseman, Oct 13 2018

nonn

			The sequence of terms together with their multiset multisystems begins:
      1: {}
     13: {{1,2}}
    113: {{1,2,3}}
    377: {{1,2},{1,3}}
    611: {{1,2},{2,3}}
   1291: {{1,2,3,4}}
   1363: {{1,3},{2,3}}
   1469: {{1,2},{1,2,3}}
   1937: {{1,2},{3,4}}
   2021: {{1,4},{2,3}}
   2117: {{1,3},{2,4}}
   3277: {{1,3},{1,2,3}}
   4537: {{1,2},{1,3,4}}
   4859: {{1,4},{1,2,3}}
   5249: {{1,3},{1,2,4}}
   5311: {{2,3},{1,2,3}}
   7423: {{1,2},{2,3,4}}
   8249: {{2,4},{1,2,3}}
   8507: {{2,3},{1,2,4}}
  16211: {{1,2},{1,3},{1,4}}

Wikipedia, Hypergraph

Cf. A003963, A005117, A055932, A056239, A112798, A255906, A290103, A302242, A302478, A305052.

Cf. A320456, A320458, A320464, A320532, A320533.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
normQ[sys_]:=Or[Length[sys]==0,Union@@sys==Range[Max@@Max@@sys]];
Select[Range[10000],And[SquareFreeQ[#],normQ[primeMS/@primeMS[#]],And@@(And[SquareFreeQ[#],PrimeOmega[#]>1]&/@primeMS[#])]&]

cp's OEIS Frontend

A320456 Numbers whose multiset multisystem spans an initial interval of positive integers.

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A320922 Heinz numbers of graphical partitions.

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A004251 Number of graphical partitions (degree-vectors for simple graphs with n vertices, or possible ordered row-sum vectors for a symmetric 0-1 matrix with diagonal values 0).

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A320461 MM-numbers of labeled graphs with loops spanning an initial interval of positive integers.

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A309356 MM-numbers of labeled simple covering graphs.

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A095268 Number of distinct degree sequences among all n-vertex graphs with no isolated vertices.

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A322551 Primes indexed by squarefree semiprimes.

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PARI

A320462 MM-numbers of labeled multigraphs with loops spanning an initial interval of positive integers.

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