A253317 -id:A253317 - cp's OEIS frontend

A368109 Number of ways to choose a binary index of each binary index of n.

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 4, 4, 4, 4, 8, 8, 8, 8, 3, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 3, 6, 6, 6, 6, 6, 6, 6, 6, 12, 12, 12

Offset: 0

Gus Wiseman, Dec 12 2023

nonn

First differs from A367912 at a(52) = 8, A367912(52) = 7.
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}.
Run-lengths are all 4 or 8.

			The binary indices of binary indices of 20 are {{1,2},{1,3}}, with choices (1,1), (1,3), (2,1), (2,3), so a(20) = 4.
The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with choices (1,1,1), (1,1,3), (1,3,2), (1,3,3), (2,1,2), (2,1,3), (2,3,2), (2,3,3), so a(52) = 8.

All entries appear to belong to A003586.
Positions of ones are A253317.
The version for prime indices is A355741, for multisets A355744.
Choosing a multiset (not sequence) gives A367912, firsts A367913.
Positions of first appearances are A368111, sorted 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, A309326, A326031, A326702, A326753, A355731, A355739, A367771, A367905, A367906, A367915.

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

a(n) = Product_{k in A048793(n)} A000120(k).

A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 7, 7, 7, 7, 4, 4, 4, 4, 7, 7, 7, 7, 3, 3, 3, 3, 5, 5, 5, 5, 3, 3, 3, 3, 5, 5, 5, 5, 5, 5, 5, 5, 8, 8, 8, 8

Offset: 0

Gus Wiseman, Dec 12 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 run-lengths are all 4 or 8.

			The binary indices of binary indices of 52 are {{1,2},{1,3},{2,3}}, with multiset choices {1,1,2}, {1,1,3}, {1,2,2}, {1,2,3}, {1,3,3}, {2,2,3}, {2,3,3}, so a(52) = 7.

Positions of ones are A253317.
The version for multisets and divisors is A355733, for sequences A355731.
The version for multisets is A355744, for sequences A355741.
For a sequence of distinct choices we have A367905, firsts A367910.
Positions of first appearances are A367913, sorted A367915.
Choosing a sequence instead of multiset gives A368109, firsts A368111.
Choosing a set instead of multiset gives A368183, firsts 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, A309326, A326031, A326702, A326753, A355735, A355739, A355740, A355745, A367771, A367906.

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

A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 3, 3, 4, 5, 6, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 5, 5, 7, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 6, 7, 6, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7, 7

Offset: 0

Gus Wiseman, Aug 19 2019

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. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18.

			22 is the BII-number of {{2},{1,2},{1,3}}, and 7 has binary indices {1,2,3}, so a(22) = 7.

Tilman Piesk, Illustration of the first 128 terms

Indices of records are A253317.

Cf. A000120, A001511, A029931, A035327, A048793, A070939, A072639, A326031, A326702, A326947, A261283 (XOR equivalent).

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Table[Total[2^Union@@bpe/@bpe[n]]/2,{n,0,100}]

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[[#]]]&]

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.

Original entry on oeis.org

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[[#]]]&]

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[#]&]

A371449 Numbers whose prime indices are not powers of 2.

Original entry on oeis.org

1, 5, 11, 13, 17, 23, 25, 29, 31, 37, 41, 43, 47, 55, 59, 61, 65, 67, 71, 73, 79, 83, 85, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 137, 139, 143, 145, 149, 151, 155, 157, 163, 167, 169, 173, 179, 181, 185, 187, 191, 193, 197, 199, 205, 211, 215

Offset: 1

Gus Wiseman, Mar 31 2024

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 terms together with their prime indices begin:
     1: {}        85: {3,7}      169: {6,6}     253: {5,9}
     5: {3}       89: {24}       173: {40}      257: {55}
    11: {5}       97: {25}       179: {41}      263: {56}
    13: {6}      101: {26}       181: {42}      269: {57}
    17: {7}      103: {27}       185: {3,12}    271: {58}
    23: {9}      107: {28}       187: {5,7}     275: {3,3,5}
    25: {3,3}    109: {29}       191: {43}      277: {59}
    29: {10}     113: {30}       193: {44}      281: {60}
    31: {11}     115: {3,9}      197: {45}      283: {61}
    37: {12}     121: {5,5}      199: {46}      289: {7,7}
    41: {13}     125: {3,3,3}    205: {3,13}    293: {62}
    43: {14}     127: {31}       211: {47}      295: {3,17}
    47: {15}     137: {33}       215: {3,14}    299: {6,9}
    55: {3,5}    139: {34}       221: {6,7}     305: {3,18}
    59: {17}     143: {5,6}      223: {48}      307: {63}
    61: {18}     145: {3,10}     227: {49}      313: {65}
    65: {3,6}    149: {35}       229: {50}      317: {66}
    67: {19}     151: {36}       233: {51}      319: {5,10}
    71: {20}     155: {3,11}     235: {3,15}    325: {3,3,6}
    73: {21}     157: {37}       239: {52}      331: {67}
    79: {22}     163: {38}       241: {53}      335: {3,19}
    83: {23}     167: {39}       251: {54}      337: {68}

Partitions of this type are counted by A101417.
For binary indices instead of prime indices we have A326781.
Requiring powers of two gives A318400, for binary indices A253317.
An opposite version is A371443.
For primes instead of powers of 2 we have A320628.
A000040 lists prime numbers, complement A018252.
A000961 lists prime-powers.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A057716 lists non-powers of 2.
A070939 gives length of binary expansion.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
Cf. A005117, A326782, A371451, A371444.

Select[Range[100],And@@Not/@IntegerQ/@Log[2, PrimePi/@First/@FactorInteger[#]]&]

cp's OEIS Frontend

A368109 Number of ways to choose a binary index of each binary index of n.

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A367912 Number of multisets that can be obtained by choosing a binary index of each binary index of n.

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A327041 a(n) is the number whose binary indices are the union of the set-system with BII-number n.

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

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

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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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A371443 Numbers whose binary indices are nonprime numbers.

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A371444 Numbers whose binary indices are composite numbers.

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