A329559
MM-numbers of multiset clutters (connected weak antichains of multisets).
Original entry on oeis.org
1, 2, 3, 5, 7, 9, 11, 13, 17, 19, 23, 25, 27, 29, 31, 37, 41, 43, 47, 49, 53, 59, 61, 67, 71, 73, 79, 81, 83, 89, 91, 97, 101, 103, 107, 109, 113, 121, 125, 127, 131, 137, 139, 149, 151, 157, 163, 167, 169, 173, 179, 181, 191, 193, 197, 199, 203, 211, 223, 227
Offset: 1
The sequence of terms tother with their corresponding clutters begins:
1: {} 37: {{1,1,2}} 91: {{1,1},{1,2}}
2: {{}} 41: {{6}} 97: {{3,3}}
3: {{1}} 43: {{1,4}} 101: {{1,6}}
5: {{2}} 47: {{2,3}} 103: {{2,2,2}}
7: {{1,1}} 49: {{1,1},{1,1}} 107: {{1,1,4}}
9: {{1},{1}} 53: {{1,1,1,1}} 109: {{10}}
11: {{3}} 59: {{7}} 113: {{1,2,3}}
13: {{1,2}} 61: {{1,2,2}} 121: {{3},{3}}
17: {{4}} 67: {{8}} 125: {{2},{2},{2}}
19: {{1,1,1}} 71: {{1,1,3}} 127: {{11}}
23: {{2,2}} 73: {{2,4}} 131: {{1,1,1,1,1}}
25: {{2},{2}} 79: {{1,5}} 137: {{2,5}}
27: {{1},{1},{1}} 81: {{1},{1},{1},{1}} 139: {{1,7}}
29: {{1,3}} 83: {{9}} 149: {{3,4}}
31: {{5}} 89: {{1,1,1,2}} 151: {{1,1,2,2}}
Cf.
A056239,
A112798,
A289509,
A302242,
A302494,
A304716,
A318991,
A319837,
A320275,
A320456,
A328514,
A329553,
A329555.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[100],And[stableQ[primeMS[#],Divisible],Length[zsm[primeMS[#]]]<=1]&]
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
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:
The nonprime instead of squarefree semiprime version:
The semiprime instead of squarefree semiprime version:
A002100 counts partitions into squarefree 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.
A320911 lists products of squarefree semiprimes (Heinz numbers of
A338914).
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).
-
sqfsemiQ[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[1000],FreeQ[If[#==1,{},FactorInteger[#]],{p_,k_}/;!sqfsemiQ[PrimePi[p]]]&]
A382077
Number of integer partitions of n that can be partitioned into a set of sets.
Original entry on oeis.org
1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 171, 217, 283, 361, 449, 574, 721, 900, 1126, 1397, 1731, 2143, 2632, 3223, 3961, 4825, 5874, 7131, 8646, 10452, 12604, 15155, 18216, 21826, 26108, 31169, 37156, 44202, 52492, 62233, 73676, 87089, 102756, 121074
Offset: 0
For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
The a(1) = 1 through a(8) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7) (8)
(2,1) (3,1) (3,2) (4,2) (4,3) (5,3)
(2,1,1) (4,1) (5,1) (5,2) (6,2)
(2,2,1) (3,2,1) (6,1) (7,1)
(3,1,1) (4,1,1) (3,2,2) (3,3,2)
(2,2,1,1) (3,3,1) (4,2,2)
(4,2,1) (4,3,1)
(5,1,1) (5,2,1)
(3,2,1,1) (6,1,1)
(3,2,2,1)
(3,3,1,1)
(4,2,1,1)
(3,2,1,1,1)
Factorizations of this type are counted by
A050345.
Normal multiset partitions of this type are counted by
A116539.
The MM-numbers of these multiset partitions are
A302494.
Twice-partitions of this type are counted by
A358914.
For distinct block-sums instead of blocks we have
A381992, ranked by
A382075.
For normal multisets instead of integer partitions we have
A382214, complement
A292432.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]
A382078
Number of integer partitions of n that cannot be partitioned into a set of sets.
Original entry on oeis.org
0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 126, 168, 207, 266, 343, 428, 534, 675, 832, 1039, 1279, 1575, 1933, 2381, 2881, 3524, 4269, 5179, 6237, 7525, 9033, 10860, 12969, 15512, 18475, 22005, 26105, 30973, 36642, 43325, 51078, 60184, 70769, 83152
Offset: 0
The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
{{1},{1,2},{1,2}}
{{1},{1},{2},{1,2}}
{{1},{1},{1},{2},{2}}
But none of these is itself a set, so y is counted under a(7).
The a(2) = 1 through a(8) = 9 partitions:
(11) (111) (22) (2111) (33) (2221) (44)
(1111) (11111) (222) (4111) (2222)
(3111) (22111) (5111)
(21111) (31111) (22211)
(111111) (211111) (41111)
(1111111) (221111)
(311111)
(2111111)
(11111111)
The MM-numbers of these multiset partitions (set of sets) are
A302494.
Twice-partitions of this type are counted by
A358914.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions into distinct sets, complement
A050345.
A265947 counts refinement-ordered pairs of integer partitions.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]
A382075
Numbers whose prime indices can be partitioned into a set of sets with distinct sums.
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84
Offset: 1
The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets with distinct sums, so 1080 is in the sequence.
Twice-partitions of this type are counted by
A279785, see also
A358914.
Normal multiset partitions into sets with distinct sums are counted by
A381718.
Partitions of this type are counted by
A381992.
For distinct blocks instead of block-sums we have
A382200, complement
A293243.
MM-numbers of multiset partitions into sets with distinct sums are
A382201.
Normal multisets of this type are counted by
A382216, see also
A382214.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
-
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]& /@ sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]& /@ sps[Range[Length[set]]]];
Select[Range[100],Length[Select[mps[prix[#]], And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]
A382200
Numbers that can be written as a product of distinct squarefree numbers.
Original entry on oeis.org
1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84
Offset: 1
The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets, so 1080 is in the sequence.
We have 18000 = 2*5*6*10*30, so 18000 is in the sequence.
Twice-partitions of this type are counted by
A279785, see also
A358914.
Normal multisets not of this type are counted by
A292432, strong
A292444.
The case of a unique choice is
A293511.
MM-numbers of multiset partitions into distinct sets are
A302494.
For distinct block-sums instead of blocks we have
A382075, counted by
A381992.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
-
N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
if numtheory:-issqrfree(n) then
S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
fi;
od:
remove(t -> A[t]=0, [$1..N]); # Robert Israel, Apr 21 2025
-
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[sqfacs[#]]>0&]
A328514
MM-numbers of connected sets of sets.
Original entry on oeis.org
1, 2, 3, 5, 11, 13, 17, 29, 31, 39, 41, 43, 47, 59, 65, 67, 73, 79, 83, 87, 101, 109, 113, 127, 129, 137, 139, 149, 157, 163, 167, 179, 181, 191, 195, 199, 211, 233, 235, 237, 241, 257, 269, 271, 277, 283, 293, 303, 313, 317, 319, 331, 339, 347, 349, 353, 365
Offset: 1
The sequence all connected set of sets together with their MM-numbers begins:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
11: {{3}}
13: {{1,2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
39: {{1},{1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
59: {{7}}
65: {{2},{1,2}}
67: {{8}}
73: {{2,4}}
79: {{1,5}}
83: {{9}}
87: {{1},{1,3}}
The not-necessarily-connected case is
A302494.
BII-numbers of connected set-systems are
A326749.
MM-numbers of connected sets of multisets are
A328513.
Cf.
A005117,
A007947,
A056239,
A112798,
A286518,
A302242,
A302569,
A304714,
A305078,
A305079,
A327398.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
Select[Range[1000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&]
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
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)
These primes (of semiprime index) are listed by
A106349.
The strict (squarefree) case is
A340020.
The prime instead of semiprime version:
The nonprime instead of semiprime version:
The squarefree semiprime instead of semiprime version:
A006881 lists squarefree semiprimes.
A037143 lists primes and semiprimes (and 1).
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).
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).
-
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]]]&]
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
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[#])]&]
A382201
MM-numbers of sets of sets with distinct sums.
Original entry on oeis.org
1, 2, 3, 5, 6, 10, 11, 13, 15, 17, 22, 26, 29, 30, 31, 33, 34, 39, 41, 43, 47, 51, 55, 58, 59, 62, 65, 66, 67, 73, 78, 79, 82, 83, 85, 86, 87, 93, 94, 101, 102, 109, 110, 113, 118, 123, 127, 129, 130, 134, 137, 139, 141, 145, 146, 149, 155, 157, 158, 163, 165
Offset: 1
The terms together with their prime indices of prime indices begin:
1: {}
2: {{}}
3: {{1}}
5: {{2}}
6: {{},{1}}
10: {{},{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
22: {{},{3}}
26: {{},{1,2}}
29: {{1,3}}
30: {{},{1},{2}}
31: {{5}}
33: {{1},{3}}
34: {{},{4}}
39: {{1},{1,2}}
Set partitions of this type are counted by
A275780.
Twice-partitions of this type are counted by
A279785.
For just sets of sets we have
A302478.
For distinct blocks instead of block-sums we have
A302494.
For equal instead of distinct sums we have
A302497.
For just distinct sums we have
A326535.
Cf.
A000720,
A003963,
A005117,
A007716,
A293511,
A302242,
A319899,
A326534,
A368100,
A368101,
A381635,
A382215.
-
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],And@@SquareFreeQ/@prix[#]&&UnsameQ@@Total/@prix/@prix[#]&]
Showing 1-10 of 36 results.
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