A320461 -id:A320461 - cp's OEIS frontend

A322661 Number of graphs with loops spanning n labeled vertices.

1, 1, 5, 45, 809, 28217, 1914733, 254409765, 66628946641, 34575388318705, 35680013894626133, 73392583417010454429, 301348381381966079690489, 2471956814761996896091805993, 40530184362443281653842556898237, 1328619783326799871943604598592805525

Offset: 0

Gus Wiseman, Dec 22 2018

nonn

The span of a graph is the union of its edges.

			The a(2) = 5 edge-sets:
  {{1,2}}
  {{1,1},{1,2}}
  {{1,1},{2,2}}
  {{1,2},{2,2}}
  {{1,1},{1,2},{2,2}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A000666, A006125, A006129 (loops not allowed), A054921, A062740, A116539, A320461, A322635, A048291 (for directed edgs).

Table[Sum[(-1)^(n-k)*Binomial[n,k]*2^Binomial[k+1,2],{k,0,n}],{n,10}]
(* second program *)
Table[Select[Expand[Product[1+x[i]*x[j],{j,n},{i,j}]],And@@Table[!FreeQ[#,x[i]],{i,n}]&]/.x[_]->1,{n,7}]

a(n) = sum(k=0, n, (-1)^(n-k)*binomial(n,k)*2^binomial(k+1,2)) \\ Andrew Howroyd, Jan 06 2024

Exponential transform of A062740, if we assume A062740(1) = 1.

Inverse binomial transform of A006125(n+1) = 2^binomial(n+1,2).

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

Original entry on oeis.org

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[#]]&]

A322700 Number of unlabeled graphs with loops spanning n vertices.

Original entry on oeis.org

1, 1, 4, 14, 70, 454, 4552, 74168, 2129348, 111535148, 10812483376, 1945437208224, 650378721156736, 404749938336404704, 470163239887698967104, 1022592854829028311090816, 4177826139658552046627175072, 32163829440870460348768023969632

Offset: 0

Gus Wiseman, Dec 23 2018

nonn

The span of a graph is the union of its edges. The not necessarily spanning case is A000666.

Andrew Howroyd, Table of n, a(n) for n = 0..80
Gus Wiseman, Non-isomorphic representatives of the a(4) = 70 spanning graphs with loops.

Cf. A000666, A006125, A006129, A054921, A062740, A320461, A322635, A322661.

Table[Sum[2^PermutationCycles[Ordering[Map[Sort,Select[Tuples[Range[n],2],OrderedQ]/.Rule@@@Table[{i,prm[[i]]},{i,n}],{1}]],Length],{prm,Permutations[Range[n]]}]/n!,{n,0,8}]//Differences (* Mathematica 8.0+ *)

from itertools import combinations
from math import prod, factorial, gcd
from fractions import Fraction
from sympy.utilities.iterables import partitions
def A322700(n): return int(sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n))-sum(Fraction(1<>1)+1)*r+(q*r*(r-1)>>1) for q, r in p.items()),prod(q**r*factorial(r) for q, r in p.items())) for p in partitions(n-1))) if n else 1 # Chai Wah Wu, Jul 14 2024

First differences of A000666.

A339655 Number of non-loop-graphical integer partitions of 2n.

Original entry on oeis.org

0, 0, 1, 3, 7, 14, 28, 51, 91, 156, 260, 425, 680, 1068, 1654, 2524, 3802, 5668, 8350, 12190, 17634, 25306, 36011, 50902, 71441, 99642

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with equal source and target. See A339657 for the Heinz numbers, and A339656 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the prime factors of n can be partitioned into distinct pairs;
(2) n can be factored into distinct semiprimes;
(3) the prime signature of n is loop-graphical.

			The a(2) = 1 through a(5) = 14 partitions (A = 10):
  (4)  (6)    (8)      (A)
       (4,2)  (4,4)    (5,5)
       (5,1)  (5,3)    (6,4)
              (6,2)    (7,3)
              (7,1)    (8,2)
              (5,2,1)  (9,1)
              (6,1,1)  (5,3,2)
                       (5,4,1)
                       (6,2,2)
                       (6,3,1)
                       (7,2,1)
                       (8,1,1)
                       (6,2,1,1)
                       (7,1,1,1)
For example, the seven normal loop-multigraphs with degrees y = (5,3,2) are:
  {{1,1},{1,1},{1,2},{2,2},{3,3}}
  {{1,1},{1,1},{1,2},{2,3},{2,3}}
  {{1,1},{1,1},{1,3},{2,2},{2,3}}
  {{1,1},{1,2},{1,2},{1,2},{3,3}}
  {{1,1},{1,2},{1,2},{1,3},{2,3}}
  {{1,1},{1,2},{1,3},{1,3},{2,2}}
  {{1,2},{1,2},{1,2},{1,3},{1,3}},
but since none of these is a loop-graph (because they are not strict), y is counted under a(5).

Eric Weisstein's World of Mathematics, Graphical partition.

A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A062740 counts labeled connected loop-graphs.
A101048 counts partitions into semiprimes.
A320461 ranks normal loop-graphs.
A322661 counts covering loop-graphs.
A320655 counts factorizations into semiprimes.
The following count vertex-degree partitions and give their Heinz numbers:
- A058696 counts partitions of 2n (A300061).
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 (this sequence) counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A007717, A025065, A147878, A320732, A320921, A338898, A338902, A338912, A338913, A339659, A339660, A339662.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]=={}&]],{n,0,5}]

A058696(n) = a(n) + A339656(n).

a(7)-a(25) from Andrew Howroyd, Jan 10 2024

A339656 Number of loop-graphical integer partitions of 2n.

Original entry on oeis.org

1, 2, 4, 8, 15, 28, 49, 84, 140, 229, 367, 577, 895, 1368, 2064, 3080, 4547, 6642, 9627, 13825, 19704, 27868, 39164, 54656, 75832, 104584

Offset: 0

Gus Wiseman, Dec 14 2020

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.
The following are equivalent characteristics for any positive integer n:
(1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;
(2) n can be factored into distinct semiprimes;
(3) the unordered prime signature of n is loop-graphical.

			The a(0) = 1 through a(4) = 15 partitions:
  ()  (2)    (2,2)      (3,3)          (3,3,2)
      (1,1)  (3,1)      (2,2,2)        (4,2,2)
             (2,1,1)    (3,2,1)        (4,3,1)
             (1,1,1,1)  (4,1,1)        (2,2,2,2)
                        (2,2,1,1)      (3,2,2,1)
                        (3,1,1,1)      (3,3,1,1)
                        (2,1,1,1,1)    (4,2,1,1)
                        (1,1,1,1,1,1)  (5,1,1,1)
                                       (2,2,2,1,1)
                                       (3,2,1,1,1)
                                       (4,1,1,1,1)
                                       (2,2,1,1,1,1)
                                       (3,1,1,1,1,1)
                                       (2,1,1,1,1,1,1)
                                       (1,1,1,1,1,1,1,1)
For example, there are four possible loop-graphs with degrees y = (2,2,1,1), namely
  {{1,1},{2,2},{3,4}}
  {{1,1},{2,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1,2},{1,4},{2,3}}
  {{1,3},{1,4},{2,2}},
so y is counted under a(3). On the other hand, there are two possible loop-multigraphs with degrees z = (4,2), namely
  {{1,1},{1,1},{2,2}}
  {{1,1},{1,2},{1,2}},
but neither of these is a loop-graph, so z is not counted under a(3).

Eric Weisstein's World of Mathematics, Graphical partition.

A339658 ranks these partitions.
A001358 lists semiprimes, with squarefree case A006881.
A006125 counts labeled graphs, with covering case A006129.
A027187 counts partitions of even length, ranked by A028260.
A062740 counts labeled connected loop-graphs.
A320461 ranks normal loop-graphs.
A320655 counts factorizations into semiprimes.
A322353 counts factorizations into distinct semiprimes.
A322661 counts covering loop-graphs.
A339845 counts the same partitions by length, or A339844 with zeros.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A000569 counts graphical partitions (A320922).
- A058696 counts partitions of 2n (A300061).
- A209816 counts multigraphical partitions (A320924).
- A321728 is conjectured to count non-half-loop-graphical partitions of n.
- A339617 counts non-graphical partitions of 2n (A339618).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 [this sequence] counts loop-graphical partitions (A339658).
The following count partitions of even length and give their Heinz numbers:
- A027187 has no additional conditions (A028260).
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001055, A001222, A025065, A095268, A101048, A320656, A320921, A338902, A338912, A338913, A339659.

spsbin[{}]:={{}};spsbin[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsbin[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mpsbin[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@spsbin[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[strnorm[2*n],Select[mpsbin[#],UnsameQ@@#&]!={}&]],{n,0,5}]

A058696(n) = a(n) + A339655(n).

a(8)-a(25) from Andrew Howroyd, Jan 10 2024

A339113 Products of primes of squarefree semiprime index (A322551).

Original entry on oeis.org

1, 13, 29, 43, 47, 73, 79, 101, 137, 139, 149, 163, 167, 169, 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, 841, 907, 929, 937

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also MM-numbers of labeled multigraphs (without uncovered vertices). 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}}.

			The sequence of terms together with the corresponding multigraphs begins:
      1: {}               233: {{2,7}}          487: {{2,11}}
     13: {{1,2}}          257: {{3,5}}          491: {{1,15}}
     29: {{1,3}}          269: {{2,8}}          499: {{3,8}}
     43: {{1,4}}          271: {{1,10}}         559: {{1,2},{1,4}}
     47: {{2,3}}          293: {{1,11}}         577: {{1,16}}
     73: {{2,4}}          313: {{3,6}}          607: {{2,12}}
     79: {{1,5}}          347: {{2,9}}          611: {{1,2},{2,3}}
    101: {{1,6}}          373: {{1,12}}         631: {{3,9}}
    137: {{2,5}}          377: {{1,2},{1,3}}    647: {{1,17}}
    139: {{1,7}}          389: {{4,5}}          653: {{4,7}}
    149: {{3,4}}          421: {{1,13}}         673: {{1,18}}
    163: {{1,8}}          439: {{3,7}}          677: {{2,13}}
    167: {{2,6}}          443: {{1,14}}         727: {{2,14}}
    169: {{1,2},{1,2}}    449: {{2,10}}         751: {{4,8}}
    199: {{1,9}}          467: {{4,6}}          757: {{1,19}}

These primes (of squarefree semiprime index) are listed by A322551.
The strict (squarefree) case is A309356.
The prime instead of squarefree semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of squarefree semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The semiprime instead of squarefree semiprime version:
  primes: A106349
  products: A339112
  strict: A340020
A001358 lists semiprimes, with odd/even terms A046315/A100484.
A002100 counts partitions into squarefree semiprimes.
A005117 lists squarefree numbers.
A006881 lists squarefree semiprimes, with odd/even terms A046388/A100484.
A056239 gives the sum of prime indices, which are listed by A112798.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A338899/A270650/A270652 give the prime indices of squarefree semiprimes.
A339561 lists products of distinct squarefree semiprimes (ranking: A339560).
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A289509, A320461.

sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]

A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 2, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 2, 1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Antti Karttunen, Dec 06 2018

nonn

A semiprime (A001358) is a product of any two prime numbers. In the even case, these factorizations have A001222(n)/2 factors. - Gus Wiseman, Dec 31 2020

Records 1, 2, 3, 4, 5, 9, 13, 15, 17, ... occur at 1, 60, 210, 840, 1260, 4620, 27720, 30030, 69300, ...

			a(4) = 1, as there is just one way to factor 4 into distinct semiprimes, namely as {4}.
From _Gus Wiseman_, Dec 31 2020: (Start)
The a(n) factorizations for n = 60, 210, 840, 1260, 4620, 12600, 18480:
  4*15   6*35    4*6*35    4*9*35    4*15*77    4*6*15*35    4*6*10*77
  6*10   10*21   4*10*21   4*15*21   4*21*55    4*6*21*25    4*6*14*55
         14*15   4*14*15   6*10*21   4*33*35    4*9*10*35    4*6*22*35
                 6*10*14   6*14*15   6*10*77    4*9*14*25    4*10*14*33
                           9*10*14   6*14*55    4*10*15*21   4*10*21*22
                                     6*22*35    6*10*14*15   4*14*15*22
                                     10*14*33                6*10*14*22
                                     10*21*22
                                     14*15*22
(End)

Unlabeled multiset partitions of this type are counted by A007717.
The version for partitions is A112020, or A101048 without distinctness.
The non-strict version is A320655.
Positions of zeros include A320892.
Positions of nonzero terms are A320912.
The case of squarefree factors is A339661, or A320656 without distinctness.
Allowing prime factors gives A339839, or A320732 without distinctness.
A322661 counts loop-graphs, ranked by A320461.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A027187 counts partitions of even length, ranked by A028260.
A037143 lists primes and semiprimes.
A338898/A338912/A338913 give the prime indices of semiprimes.
A339846 counts even-length factorizations, with ordered version A174725.
Cf. A001221, A006125, A006129, A028260, A320893, A338915, A339841.

Table[Count[Subsets[Select[Divisors[n], PrimeOmega[#] == 2 &]], ?(Times @@ # == n &)], {n, 105}] (* _Michael De Vlieger, Dec 11 2020 *)

A322353(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((2==bigomega(d)&&(d<=m)), s += A322353(n/d, d-1))); (s)); \\ Antti Karttunen, Dec 10 2020

a(n) = Sum_{d|n} (-1)^A001222(d) * A339839(n/d). - Gus Wiseman, Dec 31 2020

A339112 Products of primes of semiprime index (A106349).

Original entry on oeis.org

1, 7, 13, 23, 29, 43, 47, 49, 73, 79, 91, 97, 101, 137, 139, 149, 161, 163, 167, 169, 199, 203, 227, 233, 257, 269, 271, 293, 299, 301, 313, 329, 343, 347, 373, 377, 389, 421, 439, 443, 449, 467, 487, 491, 499, 511, 529, 553, 559, 577, 607, 611, 631, 637, 647

Offset: 1

Gus Wiseman, Mar 12 2021

nonn

A semiprime (A001358) is a product of any two prime numbers.

Also MM-numbers of labeled multigraphs with loops (without uncovered vertices). 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}}.

			The sequence of terms together with the corresponding multigraphs begins (A..F = 10..15):
     1:            149:   (34)     313:     (36)
     7:   (11)     161: (11)(22)   329:   (11)(23)
    13:   (12)     163:   (18)     343: (11)(11)(11)
    23:   (22)     167:   (26)     347:     (29)
    29:   (13)     169: (12)(12)   373:     (1C)
    43:   (14)     199:   (19)     377:   (12)(13)
    47:   (23)     203: (11)(13)   389:     (45)
    49: (11)(11)   227:   (44)     421:     (1D)
    73:   (24)     233:   (27)     439:     (37)
    79:   (15)     257:   (35)     443:     (1E)
    91: (11)(12)   269:   (28)     449:     (2A)
    97:   (33)     271:   (1A)     467:     (46)
   101:   (16)     293:   (1B)     487:     (2B)
   137:   (25)     299: (12)(22)   491:     (1F)
   139:   (17)     301: (11)(14)   499:     (38)

Robert Israel, Table of n, a(n) for n = 1..10000

These primes (of semiprime index) are listed by A106349.
The strict (squarefree) case is A340020.
The prime instead of semiprime version:
  primes: A006450
  products: A076610
  strict: A302590
The nonprime instead of semiprime version:
  primes: A007821
  products: A320628
  odd: A320629
  strict: A340104
  odd strict: A340105
The squarefree semiprime instead of semiprime version:
  strict: A309356
  primes: A322551
  products: A339113
A001358 lists semiprimes, with odd and even terms A046315 and A100484.
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
A056239 gives the sum of prime indices, which are listed by A112798.
A084126 and A084127 give the prime factors of semiprimes.
A101048 counts partitions into semiprimes.
A302242 is the weight of the multiset of multisets with MM-number n.
A305079 is the number of connected components for MM-number n.
A320892 lists even-omega non-products of distinct semiprimes.
A320911 lists products of squarefree semiprimes (Heinz numbers of A338914).
A320912 lists products of distinct semiprimes (Heinz numbers of A338916).
A338898, A338912, and A338913 give the prime indices of semiprimes.
MM-numbers: A255397 (normal), A302478 (set multisystems), A320630 (set multipartitions), A302494 (sets of sets), A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A328514 (connected sets of sets), A329559 (clutters), A340019 (half-loop graphs).
Cf. A000040, A000720, A001055, A001222, A003963, A005117, A007097, A289509, A320461.

N:= 1000: # for terms up to N
SP:= {}: p:= 1:
for i from 1 do
  p:= nextprime(p);
  if 2*p > N then break fi;
  Q:= map(t -> p*t, select(isprime, {2,seq(i,i=3..min(p,N/p),2)}));
  SP:= SP union Q;
od:
SP:= sort(convert(SP,list)):
PSP:= map(ithprime,SP):
R:= {1}:
for p in PSP do
  Rp:= {}:
  for k from 1 while p^k <= N do
    Rpk:= select(`<=`,R, N/p^k);
    Rp:= Rp union map(`*`,Rpk, p^k);
  od;
  R:= R union Rp;
od:
sort(convert(R,list)); # Robert Israel, Nov 03 2024

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

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.

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}}. This sequence lists all MM-numbers of non-loop edges.

			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

A340019 MM-numbers of labeled graphs with half-loops, without isolated vertices.

Original entry on oeis.org

1, 3, 5, 11, 13, 15, 17, 29, 31, 33, 39, 41, 43, 47, 51, 55, 59, 65, 67, 73, 79, 83, 85, 87, 93, 101, 109, 123, 127, 129, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 187, 191, 195, 199, 201, 205, 211, 215, 219, 221, 233, 235, 237, 241, 249

Offset: 1

Gus Wiseman, Jan 02 2021

nonn

Here a half-loop is an edge with only one vertex, to be distinguished from a full loop, which has two equal vertices.
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 either themselves prime or a squarefree semiprime (A006881).

			The sequence of terms together with their corresponding multisets of multisets (edge sets) begins:
     1: {}              55: {{2},{3}}      137: {{2,5}}
     3: {{1}}           59: {{7}}          139: {{1,7}}
     5: {{2}}           65: {{2},{1,2}}    141: {{1},{2,3}}
    11: {{3}}           67: {{8}}          143: {{3},{1,2}}
    13: {{1,2}}         73: {{2,4}}        145: {{2},{1,3}}
    15: {{1},{2}}       79: {{1,5}}        149: {{3,4}}
    17: {{4}}           83: {{9}}          155: {{2},{5}}
    29: {{1,3}}         85: {{2},{4}}      157: {{12}}
    31: {{5}}           87: {{1},{1,3}}    163: {{1,8}}
    33: {{1},{3}}       93: {{1},{5}}      165: {{1},{2},{3}}
    39: {{1},{1,2}}    101: {{1,6}}        167: {{2,6}}
    41: {{6}}          109: {{10}}         177: {{1},{7}}
    43: {{1,4}}        123: {{1},{6}}      179: {{13}}
    47: {{2,3}}        127: {{11}}         187: {{3},{4}}
    51: {{1},{4}}      129: {{1},{1,4}}    191: {{14}}

The version with full loops covering an initial interval is A320461.
The case covering an initial interval is A340018.
The version with full loops is A340020.
A006450 lists primes of prime index.
A106349 lists primes of semiprime index.
A257994 counts prime prime indices.
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.
A322551 lists primes of squarefree semiprime index.
A330944 counts nonprime prime indices.
A339112 lists MM-numbers of multigraphs with loops.
A339113 lists MM-numbers of multigraphs.
Cf. A000040, A000720, A001222, A005117, A056239, A076610, A112798, A289509, A302590, A305079, A326788.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],And[SquareFreeQ[#],And@@(PrimeQ[#]||(SquareFreeQ[#]&&PrimeOmega[#]==2)&/@primeMS[#])]&]

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A339655 Number of non-loop-graphical integer partitions of 2n.

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A339656 Number of loop-graphical integer partitions of 2n.

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A339113 Products of primes of squarefree semiprime index (A322551).

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A322353 Number of factorizations of n into distinct semiprimes; a(1) = 1 by convention.

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