A368109 -id:A368109 - cp's OEIS frontend

A368184 Least k such that there are exactly n ways to choose a set consisting of a different binary index of each binary index of k.

7, 1, 4, 20, 276, 320, 1088, 65856, 66112, 66624, 263232

Offset: 0

Gus Wiseman, Dec 18 2023

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:
      7: {{1},{2},{1,2}}
      1: {{1}}
      4: {{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

For strict sequences: A367910, firsts of A367905, sorted A367911.
For multisets w/o distinctness: A367913, firsts of A367912, sorted A367915.
For sequences w/o distinctness: A368111, firsts of A368109, sorted A368112.
Positions of first appearances in A368183.
The sorted version is 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, A253317, A326031, A326702, A326753, A355739, A355741, A367771, A367906, A367907.

nn=10000;
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
q=Table[Length[Union[Sort/@Select[Tuples[bpe/@bpe[n]], UnsameQ@@#&]]],{n,nn}];
k=Max@@Select[Range[Max@@q], SubsetQ[q,Range[#]]&]
Table[Position[q,n][[1,1]],{n,0,k}]

A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

Original entry on oeis.org

1, 4, 7, 20, 276, 320, 1088, 65856, 66112, 66624

Offset: 1

Gus Wiseman, Dec 18 2023

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}}
      7: {{1},{2},{1,2}}
     20: {{1,2},{1,3}}
    276: {{1,2},{1,3},{1,4}}
    320: {{1,2,3},{1,4}}
   1088: {{1,2,3},{1,2,4}}
  65856: {{1,2,3},{1,4},{1,5}}
  66112: {{1,2,3},{2,4},{1,5}}
  66624: {{1,2,3},{1,2,4},{1,5}}

For sequences we have A367911, unsorted A367910, firsts of A367905.
Multisets w/o distinctness: A367915, unsorted A367913, firsts of A367912.
Sequences w/o distinctness: A368112, unsorted A368111, firsts of A368109.
Sorted list of positions of first appearances in A368183.
The unsorted version is A368184.
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, A355739, A355741, A367771, A367906, A367907.

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

A370644 Number of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 4, 13, 13, 26, 56, 126, 243, 471, 812, 1438

Offset: 0

Gus Wiseman, Mar 11 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(7) = 13 subsets:
  .  .  .  .  .  {2,3,4,5}  {2,4,6}    {2,4,6}
                            {2,3,4,5}  {2,3,4,5}
                            {2,3,5,6}  {2,3,4,7}
                            {3,4,5,6}  {2,3,5,6}
                                       {2,3,5,7}
                                       {2,3,6,7}
                                       {2,4,5,7}
                                       {2,5,6,7}
                                       {3,4,5,6}
                                       {3,4,5,7}
                                       {3,4,6,7}
                                       {3,5,6,7}
                                       {4,5,6,7}
The a(0) = 0 through a(7) = 13 set-systems:
  .  .  .  .  .  {2}{12}{3}{13}  {2}{3}{23}       {2}{3}{23}
                                 {2}{12}{3}{13}   {2}{12}{3}{13}
                                 {12}{3}{13}{23}  {12}{3}{13}{23}
                                 {2}{12}{13}{23}  {2}{12}{13}{23}
                                                  {2}{12}{3}{123}
                                                  {2}{3}{13}{123}
                                                  {12}{3}{13}{123}
                                                  {12}{3}{23}{123}
                                                  {2}{12}{13}{123}
                                                  {2}{12}{23}{123}
                                                  {2}{13}{23}{123}
                                                  {3}{13}{23}{123}
                                                  {12}{13}{23}{123}

The version with ones allowed is A370642, minimal case of A370637.
This is the minimal case of A370643.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A367902 counts choosable set-systems, ranks A367906, unlabeled A368095.
A367903 counts non-choosable set-systems, ranks A367907, unlabeled A368094.
A370585 counts maximal choosable sets.
Cf. A072639, A140637, A326031, A355529, A367905, A368109, A370589, A370591, A370636, A370639, A370640.

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[2,n]], Select[Tuples[bpe/@#],UnsameQ@@#&]=={}&]]],{n,0,10}]

A371451 Number of connected components of the binary indices of the prime indices of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 01 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.

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 binary indices of prime indices of 805 are {{1,2},{3},{1,4}}, with 2 connected components {{1,2},{1,4}} and {{3}}, so a(805) = 2.

For prime indices of prime indices we have A305079, ones A305078.
Positions of ones are A325118.
Positions of first appearances are A325782.
For prime indices of binary indices we have A371452, ones A371291.
For binary indices of binary indices we have A326753, ones A326749.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A326964 counts connected set-systems, covering A323818.
Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s, csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[csm[bix/@prix[n]]],{n,100}]

zero_first_elem_and_bitmask_connected_elems(ys) = { my(cs = List([ys[1]]), i=1); ys[1] = 0; while(i<=#cs, for(j=2, #ys, if(ys[j]&&(0!=bitand(cs[i], ys[j])), listput(cs, ys[j]); ys[j] = 0)); i++); (ys); };
A371451(n) = if(1==n, 0, my(cs = apply(p -> primepi(p), factor(n)[, 1]~), s=0); while(#cs, cs = select(c -> c, zero_first_elem_and_bitmask_connected_elems(cs)); s++); (s)); \\ Antti Karttunen, Jan 29 2025

Data section extended to a(105) by Antti Karttunen, Jan 29 2025

A370643 Number of subsets of {2..n} such that it is not possible to choose a different binary index of each element.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 7, 23, 46, 113, 287, 680, 1546, 3374, 7191, 15008, 30016, 61013, 124354, 252577, 511229, 1031064, 2074281, 4164716, 8350912, 16729473, 33494928, 67034995, 134127390, 268325204, 536737665, 1073581062, 2147162124, 4294458549, 8589210382, 17178890873

Offset: 0

Gus Wiseman, Mar 10 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 a(0) = 0 through a(7) = 23 subsets:
  .  .  .  .  .  {2,3,4,5}  {2,4,6}      {2,4,6}
                            {2,3,4,5}    {2,3,4,5}
                            {2,3,4,6}    {2,3,4,6}
                            {2,3,5,6}    {2,3,4,7}
                            {2,4,5,6}    {2,3,5,6}
                            {3,4,5,6}    {2,3,5,7}
                            {2,3,4,5,6}  {2,3,6,7}
                                         {2,4,5,6}
                                         {2,4,5,7}
                                         {2,4,6,7}
                                         {2,5,6,7}
                                         {3,4,5,6}
                                         {3,4,5,7}
                                         {3,4,6,7}
                                         {3,5,6,7}
                                         {4,5,6,7}
                                         {2,3,4,5,6}
                                         {2,3,4,5,7}
                                         {2,3,4,6,7}
                                         {2,3,5,6,7}
                                         {2,4,5,6,7}
                                         {3,4,5,6,7}
                                         {2,3,4,5,6,7}

The case with ones allowed is A370637, differences A370589.
The minimal case is A370644, with ones A370642.
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.
Cf. A072639, A326031, A355740, A367905, A368109.
Cf. A133686, A140637, A355529, A367867, A370583, A370636, A370640.

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

More terms from Jinyuan Wang, Mar 28 2025

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 18, 20, 24, 25, 26, 30, 32, 33, 34, 35, 36, 40, 42, 45, 47, 48, 50, 51, 52, 54, 55, 60, 64, 65, 66, 68, 70, 72, 75, 78, 80, 84, 85, 86, 90, 94, 96, 99, 100, 102, 104, 105, 108, 110, 119, 120, 123, 125, 126, 127, 128, 130

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
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.
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 terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  18: {{1},{2},{2}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  25: {{1,2},{1,2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}

For prime indices of prime indices we have A320456.
For binary indices of binary indices we have A326754.
An opposite version is A371292, A371293.
The case with squarefree product of prime indices is A371448.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000],normQ[Join@@bpe/@prix[#]]&]

A371443 Numbers whose binary indices are nonprime numbers.

Original entry on oeis.org

1, 8, 9, 32, 33, 40, 41, 128, 129, 136, 137, 160, 161, 168, 169, 256, 257, 264, 265, 288, 289, 296, 297, 384, 385, 392, 393, 416, 417, 424, 425, 512, 513, 520, 521, 544, 545, 552, 553, 640, 641, 648, 649, 672, 673, 680, 681, 768, 769, 776, 777, 800, 801, 808

Offset: 1

Gus Wiseman, Mar 30 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 terms together with their binary expansions and binary indices begin:
    1:          1 ~ {1}
    8:       1000 ~ {4}
    9:       1001 ~ {1,4}
   32:     100000 ~ {6}
   33:     100001 ~ {1,6}
   40:     101000 ~ {4,6}
   41:     101001 ~ {1,4,6}
  128:   10000000 ~ {8}
  129:   10000001 ~ {1,8}
  136:   10001000 ~ {4,8}
  137:   10001001 ~ {1,4,8}
  160:   10100000 ~ {6,8}
  161:   10100001 ~ {1,6,8}
  168:   10101000 ~ {4,6,8}
  169:   10101001 ~ {1,4,6,8}
  256:  100000000 ~ {9}
  257:  100000001 ~ {1,9}
  264:  100001000 ~ {4,9}
  265:  100001001 ~ {1,4,9}
  288:  100100000 ~ {6,9}
  289:  100100001 ~ {1,6,9}
  296:  100101000 ~ {4,6,9}

For powers of 2 instead of nonprime numbers we have A253317.
For prime indices instead of binary indices we have A320628.
For prime instead of nonprime we have A326782.
For composite numbers we have A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],And@@Not/@PrimeQ/@bpe[#]&]

A371444 Numbers whose binary indices are composite numbers.

Original entry on oeis.org

8, 32, 40, 128, 136, 160, 168, 256, 264, 288, 296, 384, 392, 416, 424, 512, 520, 544, 552, 640, 648, 672, 680, 768, 776, 800, 808, 896, 904, 928, 936, 2048, 2056, 2080, 2088, 2176, 2184, 2208, 2216, 2304, 2312, 2336, 2344, 2432, 2440, 2464, 2472, 2560, 2568

Offset: 1

Gus Wiseman, Mar 30 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 terms together with their binary expansions and binary indices begin:
     8:           1000 ~ {4}
    32:         100000 ~ {6}
    40:         101000 ~ {4,6}
   128:       10000000 ~ {8}
   136:       10001000 ~ {4,8}
   160:       10100000 ~ {6,8}
   168:       10101000 ~ {4,6,8}
   256:      100000000 ~ {9}
   264:      100001000 ~ {4,9}
   288:      100100000 ~ {6,9}
   296:      100101000 ~ {4,6,9}
   384:      110000000 ~ {8,9}
   392:      110001000 ~ {4,8,9}
   416:      110100000 ~ {6,8,9}
   424:      110101000 ~ {4,6,8,9}
   512:     1000000000 ~ {10}
   520:     1000001000 ~ {4,10}
   544:     1000100000 ~ {6,10}
   552:     1000101000 ~ {4,6,10}
   640:     1010000000 ~ {8,10}
   648:     1010001000 ~ {4,8,10}
   672:     1010100000 ~ {6,8,10}

For powers of 2 instead of composite numbers we have A253317.
For prime indices we have the even case of A320628.
For prime instead of composite we have A326782.
This is the even case of A371444.
An opposite version is A371449.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A001222, A005117, A055887, A320629, A326781, A368109, A368533, A371289, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[100],EvenQ[#]&&And@@Not/@PrimeQ/@bpe[#]&]

A371448 Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.

Original entry on oeis.org

1, 2, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 26, 30, 32, 33, 34, 40, 47, 48, 51, 52, 55, 60, 64, 66, 68, 80, 85, 86, 94, 96, 102, 104, 110, 120, 123, 127, 128, 132, 136, 141, 143, 160, 165, 170, 172, 187, 188, 192, 204, 205, 208, 215, 220, 221, 226, 240, 246

Offset: 1

Gus Wiseman, Mar 31 2024

nonn

Also Heinz numbers of integer partitions whose parts have (1) squarefree product and (2) binary indices covering an initial interval.
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.
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 terms together with their binary indices of prime indices begin:
   1: {}
   2: {{1}}
   4: {{1},{1}}
   5: {{1,2}}
   6: {{1},{2}}
   8: {{1},{1},{1}}
  10: {{1},{1,2}}
  12: {{1},{1},{2}}
  15: {{2},{1,2}}
  16: {{1},{1},{1},{1}}
  17: {{1,2,3}}
  20: {{1},{1},{1,2}}
  24: {{1},{1},{1},{2}}
  26: {{1},{2,3}}
  30: {{1},{2},{1,2}}
  32: {{1},{1},{1},{1},{1}}
  33: {{2},{1,3}}
  34: {{1},{1,2,3}}
  40: {{1},{1},{1},{1,2}}
  47: {{1,2,3,4}}
  48: {{1},{1},{1},{1},{2}}
  51: {{2},{1,2,3}}

An opposite version is A371293, A371292.
Without the squarefree condition we have A371447, see also A320456, A326754.
The connected components of this multiset system are counted by A371451.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts patterns, ranked by A333217.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A131689 counts patterns by number of distinct parts.
Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[1000], SquareFreeQ[Times@@prix[#]]&&normQ[Join@@bpe/@prix[#]]&]

Intersection of A302505 and A371447.

A371453 Numbers whose binary indices are all squarefree semiprimes.

Original entry on oeis.org

32, 512, 544, 8192, 8224, 8704, 8736, 16384, 16416, 16896, 16928, 24576, 24608, 25088, 25120, 1048576, 1048608, 1049088, 1049120, 1056768, 1056800, 1057280, 1057312, 1064960, 1064992, 1065472, 1065504, 1073152, 1073184, 1073664, 1073696, 2097152, 2097184

Offset: 1

Gus Wiseman, Apr 02 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 terms together with their binary expansions and binary indices begin:
       32:                 100000 ~ {6}
      512:             1000000000 ~ {10}
      544:             1000100000 ~ {6,10}
     8192:         10000000000000 ~ {14}
     8224:         10000000100000 ~ {6,14}
     8704:         10001000000000 ~ {10,14}
     8736:         10001000100000 ~ {6,10,14}
    16384:        100000000000000 ~ {15}
    16416:        100000000100000 ~ {6,15}
    16896:        100001000000000 ~ {10,15}
    16928:        100001000100000 ~ {6,10,15}
    24576:        110000000000000 ~ {14,15}
    24608:        110000000100000 ~ {6,14,15}
    25088:        110001000000000 ~ {10,14,15}
    25120:        110001000100000 ~ {6,10,14,15}
  1048576:  100000000000000000000 ~ {21}

Partitions of this type are counted by A002100, squarefree case of A101048.
For primes instead of squarefree semiprimes we get A326782.
For prime indices instead of binary indices we have A339113, A339112.
Allowing any squarefree numbers gives A368533.
This is the squarefree case of A371454.
A001358 lists squarefree semiprimes, squarefree A006881.
A005117 lists squarefree numbers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A087086, A112798, A296119, A302478, A326031, A367905, A368109, A371450.

M:= 26: # for terms < 2^M
P:= select(isprime, [$2..(M+1)/2]): nP:= nops(P):
S:= select(`<`,{seq(seq(P[i]*P[j],i=1..j-1),j=1..nP)},M+1):
R:= map(proc(s) local i; add(2^(i-1),i=s) end proc, combinat:-powerset(S) minus {{}}):
sort(convert(R,list)); # Robert Israel, Apr 04 2024

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
sqfsemi[n_]:=SquareFreeQ[n]&&PrimeOmega[n]==2;
Select[Range[10000],And@@sqfsemi/@bix[#]&]

def A371453(n): return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1')

from math import isqrt
from sympy import primepi, primerange
def A371453(n):
    def f(x,n): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    def A006881(n):
        m, k = n, f(n,n)
        while m != k:
            m, k = k, f(k,n)
        return m
    return sum(1<<A006881(i)-1 for i, j in enumerate(bin(n)[:1:-1],1) if j=='1') # Chai Wah Wu, Aug 16 2024

cp's OEIS Frontend

A368184 Least k such that there are exactly n ways to choose a set consisting of a different binary index of each binary index of k.

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A368185 Sorted list of positions of first appearances in A368183 (number of sets that can be obtained by choosing a different binary index of each binary index).

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A370644 Number of minimal subsets of {2..n} such that it is not possible to choose a different binary index of each element.

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A371451 Number of connected components of the binary indices of the prime indices of n.

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A370643 Number of subsets of {2..n} such that it is not possible to choose a different binary index of each element.

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A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

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A371443 Numbers whose binary indices are nonprime numbers.

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A371444 Numbers whose binary indices are composite numbers.

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A371448 Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.

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