A072639 -id:A072639 - cp's OEIS frontend

A367913 Least number k such that there are exactly n ways to choose a multiset consisting of a binary index of each binary index of k.

1, 4, 64, 20, 68, 320, 52, 84, 16448, 324, 832, 116, 1104, 308, 816, 340, 836, 848, 1108, 1136, 1360, 3152, 16708, 372, 5188, 5216, 852, 880, 2884, 1364, 13376, 1392, 3184, 3424, 17220, 5204, 5220, 2868, 5728, 884, 19536, 66896, 2900, 1396, 21572, 3188, 3412

Offset: 1

Gus Wiseman, Dec 16 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
      1: {{1}}
      4: {{1,2}}
     64: {{1,2,3}}
     20: {{1,2},{1,3}}
     68: {{1,2},{1,2,3}}
    320: {{1,2,3},{1,4}}
     52: {{1,2},{1,3},{2,3}}
     84: {{1,2},{1,3},{1,2,3}}
  16448: {{1,2,3},{1,2,3,4}}
    324: {{1,2},{1,2,3},{1,4}}
    832: {{1,2,3},{1,4},{2,4}}
    116: {{1,2},{1,3},{2,3},{1,2,3}}

A version for multisets and divisors is A355734.
With distinctness we have A367910, firsts of A367905, sorted A367911.
Positions of first appearances in A367912.
The sorted version is A367915.
For sequences we have A368111, firsts of A368109, sorted A368112.
For sets we have A368184, firsts of A368183, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355741, A355744.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
spnm[y_]:=Max@@NestWhile[Most,y,Union[#]!=Range[0,Max@@#]&];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,1000}];
Table[Position[c,n][[1,1]],{n,spnm[c]}]

A367915 Sorted positions of first appearances in A367912 (number of multisets that can be obtained by choosing a binary index of each binary index).

Original entry on oeis.org

1, 4, 20, 52, 64, 68, 84, 116, 308, 320, 324, 340, 372, 816, 832, 836, 848, 852, 880, 884, 1104, 1108, 1136, 1360, 1364, 1392, 1396, 1904, 1908, 2868, 2884, 2900, 2932, 3152, 3184, 3188, 3412, 3424, 3440, 3444, 3952, 3956, 5188, 5204, 5216, 5220, 5236, 5476

Offset: 1

Gus Wiseman, Dec 16 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
     1: {{1}}
     4: {{1,2}}
    20: {{1,2},{1,3}}
    52: {{1,2},{1,3},{2,3}}
    64: {{1,2,3}}
    68: {{1,2},{1,2,3}}
    84: {{1,2},{1,3},{1,2,3}}
   116: {{1,2},{1,3},{2,3},{1,2,3}}
   308: {{1,2},{1,3},{2,3},{1,4}}
   320: {{1,2,3},{1,4}}
   324: {{1,2},{1,2,3},{1,4}}
   340: {{1,2},{1,3},{1,2,3},{1,4}}
   372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}

A version for multisets and divisors is A355734.
Sorted positions of first appearances in A367912, for sequences A368109.
The unsorted version is A367913.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326749, A326753, A355733, A355744, A367905, A367906, A367911, A368112, A368185.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Union[Sort/@Tuples[bpe/@bpe[n]]]],{n,10000}];
Select[Range[Length[c]],FreeQ[Take[c,#-1],c[[#]]]&]

A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 1, 6, 17, 42, 67, 175, 400, 870, 1841, 3820, 7837, 15920, 30997, 63370, 128348, 258699, 520042, 1043284, 2090732, 4186382, 8379022, 16765549, 33540664, 67092258, 134198633, 268412631, 536844414, 1073710403, 2147296425, 4294753612, 8589686922, 17179580003

Offset: 0

Gus Wiseman, Mar 08 2024

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The binary indices of {1,4,5} are {{1},{3},{1,3}}, from which it is not possible to choose three different elements, so S is counted under a(3).
The binary indices of S = {1,6,8,9} are {{1},{2,3},{4},{1,4}}, from which it is not possible to choose four different elements, so S is counted under a(9).
The a(0) = 0 through a(6) = 17 subsets:
  .  .  .  {1,2,3}  {1,2,3,4}  {1,4,5}      {2,4,6}
                               {1,2,3,5}    {1,2,3,6}
                               {1,2,4,5}    {1,2,4,6}
                               {1,3,4,5}    {1,2,5,6}
                               {2,3,4,5}    {1,3,4,6}
                               {1,2,3,4,5}  {1,3,5,6}
                                            {1,4,5,6}
                                            {2,3,4,6}
                                            {2,3,5,6}
                                            {2,4,5,6}
                                            {3,4,5,6}
                                            {1,2,3,4,6}
                                            {1,2,3,5,6}
                                            {1,2,4,5,6}
                                            {1,3,4,5,6}
                                            {2,3,4,5,6}
                                            {1,2,3,4,5,6}

Wikipedia, Axiom of choice.

Simple graphs not of this type are counted by A133686, covering A367869.
Unlabeled graphs of this type are counted by A140637, complement A134964.
Simple graphs of this type are counted by A367867, covering A367868.
Set systems not of this type are counted by A367902, ranks A367906.
Set systems of this type are counted by A367903, ranks A367907.
Set systems uniquely not of this type are counted by A367904, ranks A367908.
Unlabeled multiset partitions of this type are A368097, complement A368098.
A version for MM-numbers of multisets is A355529, complement A368100.
Factorizations are counted by A368413/A370813, complement A368414/A370814.
The complement for prime indices is A370586, differences of A370582.
For prime indices we have A370587, differences of A370583.
Partial sums are A370637/A370643, minima A370642/A370644.
The complement is counted by A370639, partial sums A370636.
The version for a unique choice is A370641, partial sums A370638.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A058891 counts set-systems, A003465 covering, A323818 connected.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
Cf. A000612, A072639, A326702, A355739, A355740, A367772, A367905, A367909, A367912, A368094, A368095, A368109, A370640.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Select[Subsets[Range[n]],MemberQ[#,n] && Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]],{n,0,10}]

a(19)-a(35) from Alois P. Heinz, Mar 09 2024

A329557 Smallest MM-number of a set of n nonempty sets.

Original entry on oeis.org

1, 3, 15, 165, 2145, 36465, 1057485, 32782035, 1344063435, 57794727705, 2716352202135, 160264779925965, 10737740255039655, 783855038617894815, 61924548050813690385, 5139737488217536301955, 519113486309971166497455, 56583370007786857148222595, 6393920810879914857749153235

Offset: 0

Gus Wiseman, Nov 17 2019

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 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 their corresponding systems begins:
        1: {}
        3: {{1}}
       15: {{1},{2}}
      165: {{1},{2},{3}}
     2145: {{1},{2},{3},{1,2}}
    36465: {{1},{2},{3},{1,2},{4}}
  1057485: {{1},{2},{3},{1,2},{4},{1,3}}

MM-numbers of sets of sets are A302494.
MM-numbers of sets of nonempty sets are A329629.
The version allowing empty sets is A329558.
The version without singletons is A329554.
Cf. A056239, A072639, A112798, A302242, A326031, A329552, A329555, A329556.
Other MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
dae=Select[Range[10000],SquareFreeQ[#]&&And@@SquareFreeQ/@primeMS[#]&&FreeQ[primeMS[#],1]&];
Table[dae[[Position[PrimeOmega/@dae,k][[1,1]]]],{k,First[Split[Union[PrimeOmega/@dae],#2==#1+1&]]}]

a(n) = my(k=1); prod(i=1, n, until(issquarefree(k), k++); prime(k)); \\ Jinyuan Wang, Feb 23 2025

a(n) = A329558(n + 1)/2.

More terms from Jinyuan Wang, Feb 23 2025

A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

Original entry on oeis.org

1, 4, 64, 20, 68, 52, 1088, 84, 308, 1092, 116, 5184, 820, 1108, 372, 5188, 2868, 1140, 13376, 884, 5204, 17204, 1396, 13380, 2932, 5236, 275520, 19252, 1908, 13396, 17268, 5492, 275524, 84788, 3956, 13428, 1324096, 19316, 6004, 275540, 215860, 18292, 13684

Offset: 1

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   64: {{1,2,3}}
   20: {{1,2},{1,3}}
   68: {{1,2},{1,2,3}}
   52: {{1,2},{1,3},{2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

With distinctness we have A367910, sorted A367911, firsts of A367905.
For multisets we have A367913, sorted A367915, firsts of A367912.
Positions of first appearances in A368109.
The sorted version is A368112.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A309326, A326031, A326702, A326753, A355731, A355741, A355744, A367906.

nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
dd=Select[Range[nn],Max@@First/@FactorInteger[#]<=3&];
qq=Table[Length[Tuples[bpe/@bpe[n]]],{n,nn}];
kk=Select[Range[Length[dd]],SubsetQ[qq,Take[dd,#]]&]
Table[Position[qq,dd[[n]]][[1,1]],{n,kk}]

A368112 Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

Original entry on oeis.org

1, 4, 20, 52, 64, 68, 84, 116, 308, 372, 820, 884, 1088, 1092, 1108, 1140, 1396, 1908, 2868, 2932, 3956, 5184, 5188, 5204, 5236, 5492, 6004, 8052, 13376, 13380, 13396, 13428, 13684, 14196, 16244, 17204, 17268, 18292, 19252, 19316, 20340, 22388, 24436, 30580

Offset: 1

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The terms together with the corresponding set-systems begin:
    1: {{1}}
    4: {{1,2}}
   20: {{1,2},{1,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
   68: {{1,2},{1,2,3}}
   84: {{1,2},{1,3},{1,2,3}}
  116: {{1,2},{1,3},{2,3},{1,2,3}}
  308: {{1,2},{1,3},{2,3},{1,4}}
  372: {{1,2},{1,3},{2,3},{1,2,3},{1,4}}
  820: {{1,2},{1,3},{2,3},{1,4},{2,4}}
  884: {{1,2},{1,3},{2,3},{1,2,3},{1,4},{2,4}}

For multisets we have A367915, unsorted A367913, firsts A367912.
Sorted positions of first appearances in A368109.
The unsorted version is A368111.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A253317, A326031, A326702, A326753, A355741, A367771, A367905, A367906, A367911, A368184.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
c=Table[Length[Tuples[bpe/@bpe[n]]], {n,1000}];
Select[Range[Length[c]], FreeQ[Take[c,#-1],c[[#]]]&]

A370642 Number of minimal subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 1, 1, 3, 9, 26, 26, 40, 82, 175, 338, 636, 1114

Offset: 0

Gus Wiseman, Mar 10 2024

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.

			The a(0) = 0 through a(6) = 9 subsets:
  .  .  .  {1,2,3}  {1,2,3}  {1,2,3}    {1,2,3}
                             {1,4,5}    {1,4,5}
                             {2,3,4,5}  {2,4,6}
                                        {1,2,5,6}
                                        {1,3,4,6}
                                        {1,3,5,6}
                                        {2,3,4,5}
                                        {2,3,5,6}
                                        {3,4,5,6}

For prime indices we have A370591, minima of A370583, complement A370582.
This is the minimal case of A370637, complement A370636.
The version for a unique choice is A370638, maxima A370640, diffs A370641.
The case without ones is A370644.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A326031 gives weight of the set-system with BII-number n.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A368100 ranks choosable multisets, complement A355529.
A370585 counts maximal choosable sets.
Cf. A072639, A140637, A367905, A368109, A370589, A370593, A370639.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
fasmin[y_]:=Complement[y,Union@@Table[Union[s,#]& /@ Rest[Subsets[Complement[Union@@y,s]]],{s,y}]];
Table[Length[fasmin[Select[Subsets[Range[n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]]],{n,0,10}]

A368183 Number of sets that can be obtained by choosing a different binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 2, 1, 2, 1, 3, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 2, 2, 1, 1, 3, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 3, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 2, 2, 1, 3, 1, 1, 0, 2, 1, 1, 0, 1, 0, 0, 0, 3, 1, 1, 0, 1, 0, 0

Offset: 0

Gus Wiseman, Dec 17 2023

nonn

A binary index of n (row n of A048793) is any position of a 1 in its reversed binary expansion. For example, 18 has reversed binary expansion (0,1,0,0,1) and binary indices {2,5}.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,3,2), (2,1,3), both permutations of {1,2,3}, so a(52) = 1.

For sequences we have A367905, firsts A367910, sorted A367911.
Positions of zeros are A367907.
Without distinctness we have A367912, firsts A367913, sorted A367915.
Positions of positive terms are A367906.
For sequences without distinctness: A368109, firsts A368111, sorted A368112.
Positions of first appearances are A368184, sorted A368185.
A048793 lists binary indices, length A000120, sum A029931.
A058891 counts set-systems, covering A003465, connected A323818.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A072639, A309326, A326031, A326702, A326753, A355739, A355741, A355744, A367771.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]],UnsameQ@@#&]]],{n,0,100}]

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

Gus Wiseman, Nov 17 2019

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 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}}. Then a(n) is the smallest MM-number of a set of n sets.

			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}}

Jinyuan Wang, Table of n, a(n) for n = 0..100

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.
The connected version is A329552.
Cf. A005117, A048672, A056239, A072639, A112798, A302242, A326031, A329555.
Classes of MM-numbers: A305078 (connected), A316476 (antichains), A318991 (chains), A320456 (covers), A329559 (clutters).

sqvs=Select[Range[30],SquareFreeQ];
Table[Times@@Prime/@Take[sqvs,k],{k,0,Length[sqvs]}]

a(n > 0) = 2 * A329557(n - 1).

a(n) = Product_{i = 1..n} prime(A005117(i)).

a(19) from Jinyuan Wang, Feb 24 2020

A353743 Least number with run-sum trajectory of length k; a(0) = 1.

Original entry on oeis.org

1, 2, 4, 12, 84, 1596, 84588, 11081028, 3446199708, 2477817590052, 4011586678294188, 14726534696017964148, 120183249654202605411828, 2146833388573021140471483564, 83453854313999050793547980583372, 7011542477899258250521520684673165324

Offset: 0

Gus Wiseman, Jun 11 2022

nonn

Every sequence can be uniquely split into a sequence of non-overlapping runs. For example, the runs of (2,2,1,1,1,3,2,2) are ((2,2),(1,1,1),(3),(2,2)), with sums (4,3,3,4). The run-sum trajectory is obtained by repeatedly taking the run-sum transformation (A353832, A353847) until a squarefree number is reached. For example, the trajectory 12 -> 9 -> 7 corresponds to the partitions (2,1,1) -> (2,2) -> (4).

			The terms together with their prime indices begin:
      1: {}
      2: {1}
      4: {1,1}
     12: {1,1,2}
     84: {1,1,2,4}
   1596: {1,1,2,4,8}
  84588: {1,1,2,4,8,16}

The ordered version is A072639, for run-lengths A333629.
The version for run-lengths is A325278, firsts in A182850 or A323014.
The run-sum trajectory is the iteration of A353832.
The first length-k row of A353840 has index a(k).
Other sequences pertaining to this trajectory are A353841-A353846.
A001222 counts prime factors, distinct A001221.
A056239 adds up prime indices, row sums of A112798 and A296150.
A300273 ranks collapsible partitions, counted by A275870.
A353833 ranks partitions with all equal run-sums, counted by A304442.
A353835 counts distinct run-sums of prime indices, weak A353861.
A353838 ranks partitions with all distinct run-sums, counted by A353837.
A353866 ranks rucksack partitions, counted by A353864.
Cf. A002033, A005117, A006939, A071625, A076954, A126796, A181819, A182857, A188431, A299702, A325780, A325781, A353834.

Join[{1,2},Table[2*Product[Prime[2^k],{k,0,n}],{n,0,6}]]

a(n > 1) = 2 * Product_{k=0..n-2} prime(2^k).

a(n > 0) = 2 * A325782(n).

cp's OEIS Frontend

A367913 Least number k such that there are exactly n ways to choose a multiset consisting of a binary index of each binary index of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A367915 Sorted positions of first appearances in A367912 (number of multisets that can be obtained by choosing a binary index of each binary index).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370589 Number of subsets of {1..n} containing n such that it is not possible to choose a different binary index of each element.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A329557 Smallest MM-number of a set of n nonempty sets.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A368111 Least k such that there are exactly A003586(n) ways to choose a binary index of each binary index of k.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A368112 Sorted positions of first appearances in A368109 (number of ways to choose a binary index of each binary index).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A370642 Number of minimal subsets of {1..n} such that it is not possible to choose a different binary index of each element.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A368183 Number of sets that can be obtained by choosing a different binary index of each binary index of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A329558 Product of primes indexed by the first n squarefree numbers.

Views