A329552
Smallest MM-number of a connected set of n sets.
Original entry on oeis.org
1, 2, 39, 195, 5655, 62205, 2674815
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
39: {{1},{1,2}}
195: {{1},{2},{1,2}}
5655: {{1},{2},{1,2},{1,3}}
62205: {{1},{2},{3},{1,2},{1,3}}
2674815: {{1},{2},{3},{1,2},{1,3},{1,4}}
MM-numbers of connected set-systems are
A328514.
The weight of the system with MM-number n is
A302242(n).
Maximum connected divisor is
A327076.
BII-numbers of connected sets of sets are
A326749.
The smallest BII-number of a connected set of n sets is
A329625(n).
Allowing edges to have repeated vertices gives
A329553.
Requiring the edges to form an antichain gives
A329555.
The smallest MM-number of a set of n nonempty sets is
A329557(n).
Cf.
A048143,
A056239,
A112798,
A302494,
A304714,
A304716,
A305079,
A322389,
A328513,
A329554,
A329556,
A329558.
-
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]]]]]]]]];
da=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[da[[Position[PrimeOmega/@da,n][[1,1]]]],{n,First[Split[Union[PrimeOmega/@da],#2==#1+1&]]}]
A329558
Product of primes indexed by the first n squarefree numbers.
Original entry on oeis.org
1, 2, 6, 30, 330, 4290, 72930, 2114970, 65564070, 2688126870, 115589455410, 5432704404270, 320529559851930, 21475480510079310, 1567710077235789630, 123849096101627380770, 10279474976435072603910, 1038226972619942332994910, 113166740015573714296445190, 12787841621759829715498306470
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
6: {{},{1}}
30: {{},{1},{2}}
330: {{},{1},{2},{3}}
4290: {{},{1},{2},{3},{1,2}}
72930: {{},{1},{2},{3},{1,2},{4}}
2114970: {{},{1},{2},{3},{1,2},{4},{1,3}}
The smallest BII-number of a set of n sets is
A000225(n).
MM-numbers of sets of sets are
A302494.
The case without empty edges is
A329557.
The case without singletons is
A329556.
The case without empty edges or singletons is
A329554.
-
sqvs=Select[Range[30],SquareFreeQ];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]
A329554
Smallest MM-number of a set of n nonempty sets with no singletons.
Original entry on oeis.org
1, 13, 377, 16211, 761917, 55619941, 4393975339, 443791509239, 50148440544007, 6870336354528959, 954976753279525301, 142291536238649269849, 23193520406899830985387, 3873317907952271774559629, 701070541339361191195292849, 139513037726532877047863276951
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
13: {{1,2}}
377: {{1,2},{1,3}}
16211: {{1,2},{1,3},{1,4}}
761917: {{1,2},{1,3},{1,4},{2,3}}
The smallest BII-number of a set of n sets is
A000225(n).
BII-numbers of set-systems with no singletons are
A326781.
MM-numbers of sets of nonempty sets are the odd terms of
A302494.
MM-numbers of multisets of nonempty non-singleton sets are
A320629.
The version with empty edges is
A329556.
The version with singletons is
A329557.
The version with empty edges and singletons is
A329558.
-
sqvs=Select[Range[2,30],SquareFreeQ[#]&&!PrimeQ[#]&];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]
A329556
Smallest MM-number of a set of n sets with no singletons.
Original entry on oeis.org
1, 2, 26, 754, 32422, 1523834
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
26: {{},{1,2}}
754: {{},{1,2},{1,3}}
32422: {{},{1,2},{1,3},{1,4}}
1523834: {{},{1,2},{1,3},{1,4},{2,3}}
MM-numbers of sets of sets with no singletons are
A329630.
The case without empty edges is
A329554.
MM-numbers of sets of sets are
A302494.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],_?PrimeQ]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]
A329629
Products of distinct odd primes of squarefree index.
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, 113, 123, 127, 129, 137, 139, 141, 143, 145, 149, 155, 157, 163, 165, 167, 177, 179, 181, 187, 191, 195, 199, 201, 205, 211, 215, 219, 221, 233, 235, 237
Offset: 1
The sequence of terms together with their corresponding set-systems begins:
1: {}
3: {{1}}
5: {{2}}
11: {{3}}
13: {{1,2}}
15: {{1},{2}}
17: {{4}}
29: {{1,3}}
31: {{5}}
33: {{1},{3}}
39: {{1},{1,2}}
41: {{6}}
43: {{1,4}}
47: {{2,3}}
51: {{1},{4}}
55: {{2},{3}}
59: {{7}}
65: {{2},{1,2}}
67: {{8}}
73: {{2,4}}
Allowing even terms (systems with empty edges) gives
A302494.
-
Select[Range[100],OddQ[#]&&SquareFreeQ[#]&&And@@SquareFreeQ/@PrimePi/@First/@If[#==1,{},FactorInteger[#]]&]
A329553
Smallest MM-number of a connected set of n multisets.
Original entry on oeis.org
1, 2, 21, 195, 1365, 25935, 435435
Offset: 0
The sequence of terms together with their corresponding systems begins:
1: {}
2: {{}}
21: {{1},{1,1}}
195: {{1},{2},{1,2}}
1365: {{1},{2},{1,1},{1,2}}
25935: {{1},{2},{1,1},{1,2},{1,1,1}}
435435: {{1},{2},{1,1},{3},{1,2},{1,3}}
MM-numbers of connected sets of sets are
A328514.
The weight of the system with MM-number n is
A302242(n).
Maximum connected divisor is
A327076.
BII-numbers of connected set-systems are
A326749.
The smallest BII-number of a connected set-system is
A329625.
The case of strict edges is
A329552.
The smallest MM-number of a set of n nonempty sets is
A329557(n).
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
zsm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}],GCD@@s[[#]]>1&]},If[c=={},s,zsm[Sort[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
dae=Select[Range[100000],SquareFreeQ[#]&&Length[zsm[primeMS[#]]]<=1&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]
A329630
Products of distinct primes of nonprime squarefree index.
Original entry on oeis.org
1, 2, 13, 26, 29, 43, 47, 58, 73, 79, 86, 94, 101, 113, 137, 139, 146, 149, 158, 163, 167, 181, 199, 202, 226, 233, 257, 269, 271, 274, 278, 293, 298, 313, 317, 326, 334, 347, 349, 362, 373, 377, 389, 397, 398, 421, 439, 443, 449, 466, 467, 487, 491, 499, 514
Offset: 1
The sequence of terms together with their corresponding sets of sets begins:
1: {}
2: {{}}
13: {{1,2}}
26: {{},{1,2}}
29: {{1,3}}
43: {{1,4}}
47: {{2,3}}
58: {{},{1,3}}
73: {{2,4}}
79: {{1,5}}
86: {{},{1,4}}
94: {{},{2,3}}
101: {{1,6}}
113: {{1,2,3}}
137: {{2,5}}
139: {{1,7}}
146: {{},{2,4}}
149: {{3,4}}
158: {{},{1,5}}
163: {{1,8}}
MM-numbers of sets of nonempty sets are
A329629.
Products of primes of nonprime squarefree index are
A320630.
Products of prime numbers of squarefree index are
A302478.
Products of primes of nonprime index are
A320628.
-
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&!MemberQ[primeMS[#],_?PrimeQ]&]
A329661
BII-number of the set-system whose MM-number is A329629(n).
Original entry on oeis.org
0, 1, 2, 8, 4, 3, 128, 16, 32768, 9, 5, 2147483648, 256, 32, 129, 10, 9223372036854775808, 6, 170141183460469231731687303715884105728, 512, 65536, 57896044618658097711785492504343953926634992332820282019728792003956564819968, 130, 17, 32769, 4294967296
Offset: 1
The sequence of all set-systems together with their MM-numbers and BII-numbers begins:
{}: 1 ~ 0
{{1}}: 3 ~ 1
{{2}}: 5 ~ 2
{{3}}: 11 ~ 8
{{1,2}}: 13 ~ 4
{{1},{2}}: 15 ~ 3
{{4}}: 17 ~ 128
{{1,3}}: 29 ~ 16
{{5}}: 31 ~ 32768
{{1},{3}}: 33 ~ 9
{{1},{1,2}}: 39 ~ 5
{{6}}: 41 ~ 2147483648
{{1,4}}: 43 ~ 256
{{2,3}}: 47 ~ 32
{{1},{4}}: 51 ~ 129
{{2},{3}}: 55 ~ 10
{{7}}: 59 ~ 9223372036854775808
{{2},{1,2}}: 65 ~ 6
{{8}}: 67 ~ 170141183460469231731687303715884105728
{{2,4}}: 73 ~ 512
MM-numbers of set-systems are
A329629.
-
fbi[q_]:=If[q=={},0,Total[2^q]/2];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
das=Select[Range[100],OddQ[#]&&SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&];
Table[fbi[fbi/@primeMS/@primeMS[n]],{n,das}]
A371450
MM-number of the set-system with BII-number n.
Original entry on oeis.org
1, 3, 5, 15, 13, 39, 65, 195, 11, 33, 55, 165, 143, 429, 715, 2145, 29, 87, 145, 435, 377, 1131, 1885, 5655, 319, 957, 1595, 4785, 4147, 12441, 20735, 62205, 47, 141, 235, 705, 611, 1833, 3055, 9165, 517, 1551, 2585, 7755, 6721, 20163, 33605, 100815, 1363, 4089
Offset: 0
The set-system with BII-number 30 is {{2},{1,2},{3},{1,3}} with MM-number prime(3) * prime(6) * prime(5) * prime(10) = 20735.
The terms together with their prime indices and binary indices of prime indices begin:
1 -> {} -> {}
3 -> {2} -> {{1}}
5 -> {3} -> {{2}}
15 -> {2,3} -> {{1},{2}}
13 -> {6} -> {{1,2}}
39 -> {2,6} -> {{1},{1,2}}
65 -> {3,6} -> {{2},{1,2}}
195 -> {2,3,6} -> {{1},{2},{1,2}}
11 -> {5} -> {{3}}
33 -> {2,5} -> {{1},{3}}
55 -> {3,5} -> {{2},{3}}
165 -> {2,3,5} -> {{1},{2},{3}}
143 -> {5,6} -> {{1,2},{3}}
429 -> {2,5,6} -> {{1},{1,2},{3}}
715 -> {3,5,6} -> {{2},{1,2},{3}}
2145 -> {2,3,5,6} -> {{1},{2},{1,2},{3}}
A019565 gives Heinz number of binary indices.
A070939 gives length of binary expansion.
Cf.
A000720,
A003963,
A087086,
A096111,
A275024,
A302242,
A302505,
A302521,
A326782,
A329557,
A329630,
A368109.
-
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Times@@Prime/@(Times@@Prime/@#&/@bix/@bix[n]),{n,0,30}]
Showing 1-9 of 9 results.
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