A371292 -id:A371292 - cp's OEIS frontend

A371452 Number of connected components of the prime indices of the binary indices of n.

1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5

Offset: 1

Gus Wiseman, Apr 01 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.

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 prime indices of binary indices of 281492156579880 are {{1,1},{1,2},{3,4},{4,4}}, with 2 connected components {{1,1},{1,2}} and {{3,4},{4,4}}, so a(281492156579880) = 2.

Positions of first appearances are A080355, opposite A325782.
For prime indices of prime indices we have A305079, ones A305078.
For binary indices of binary indices we have A326753, ones A326749.
Positions of ones are A371291.
For binary indices of prime indices we have A371451, ones A325118.
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[prix/@bix[n]]],{n,100}]

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

A371289 Numbers whose binary indices have squarefree product.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 16, 17, 18, 19, 20, 21, 22, 23, 32, 33, 48, 49, 64, 65, 66, 67, 68, 69, 70, 71, 80, 81, 82, 83, 84, 85, 86, 87, 96, 97, 112, 113, 512, 513, 516, 517, 576, 577, 580, 581, 1024, 1025, 1026, 1027, 1028, 1029, 1030, 1031, 1040, 1041, 1042

Offset: 1

Gus Wiseman, Mar 25 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:
     0:              0 ~ {}
     1:              1 ~ {1}
     2:             10 ~ {2}
     3:             11 ~ {1,2}
     4:            100 ~ {3}
     5:            101 ~ {1,3}
     6:            110 ~ {2,3}
     7:            111 ~ {1,2,3}
    16:          10000 ~ {5}
    17:          10001 ~ {1,5}
    18:          10010 ~ {2,5}
    19:          10011 ~ {1,2,5}
    20:          10100 ~ {3,5}
    21:          10101 ~ {1,3,5}
    22:          10110 ~ {2,3,5}
    23:          10111 ~ {1,2,3,5}
    32:         100000 ~ {6}
    33:         100001 ~ {1,6}
    48:         110000 ~ {5,6}
    49:         110001 ~ {1,5,6}
    64:        1000000 ~ {7}
    65:        1000001 ~ {1,7}
    66:        1000010 ~ {2,7}

For prime instead of binary indices we have A302505.
For squarefree parts we have A368533, for prime indices A302478.
A005117 lists squarefree numbers.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
Cf. A325118, A326782, A371290, A371291, A371292, A371293, A371443, A371446, A371448, A371449, A371452, A371453.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[0,100],SquareFreeQ[Times@@bpe[#]]&]

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[#]]&]

A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

Original entry on oeis.org

1, 2, 3, 6, 7, 22, 23, 32, 33, 48, 49, 86, 87, 112, 113, 516, 517, 580, 581, 1110, 1111, 1136, 1137, 1604, 1605, 5206, 5207, 5232, 5233, 5700, 5701, 8212, 8213, 9236, 9237, 13332, 13333, 16386, 16387, 16450, 16451, 17474, 17475, 21570, 21571, 24576, 24577

Offset: 1

Gus Wiseman, Mar 28 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.

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 prime indices of binary indices begin:
    1: {{}}
    2: {{1}}
    3: {{},{1}}
    6: {{1},{2}}
    7: {{},{1},{2}}
   22: {{1},{2},{3}}
   23: {{},{1},{2},{3}}
   32: {{1,2}}
   33: {{},{1,2}}
   48: {{3},{1,2}}
   49: {{},{3},{1,2}}
   86: {{1},{2},{3},{4}}
   87: {{},{1},{2},{3},{4}}
  112: {{3},{1,2},{4}}
  113: {{},{3},{1,2},{4}}
  516: {{2},{1,3}}
  517: {{},{2},{1,3}}
  580: {{2},{4},{1,3}}
  581: {{},{2},{4},{1,3}}

Without the covering condition we have A371289.
Without squarefree product we have A371292.
Interchanging binary and prime indices gives A371448.
A000009 counts partitions covering initial interval, compositions A107429.
A000670 counts ordered set partitions, allowing empty sets A000629.
A005117 lists squarefree numbers.
A011782 counts multisets covering an initial interval.
A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
A070939 gives length of binary expansion.
A096111 gives product of binary indices.
A131689 counts patterns by number of distinct parts.
A302521 lists MM-numbers of set partitions, with empties A302505.
A326701 lists BII-numbers of set partitions.
A368533 lists numbers with squarefree binary indices, prime indices A302478.
Cf. A000040, A001222, A255906, A326782, A371291, A371294, A371447, A371452.

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

Intersection of A371292 and A371289.

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.

A371290 Numbers whose product of binary indices is a prime power > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 64, 65, 128, 129, 130, 131, 136, 137, 138, 139, 256, 257, 260, 261, 1024, 1025, 4096, 4097, 32768, 32769, 32770, 32771, 32776, 32777, 32778, 32779, 32896, 32897, 32898, 32899, 32904, 32905, 32906, 32907, 65536, 65537, 262144

Offset: 1

Gus Wiseman, Mar 27 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:
       1:                   1 ~ {1}
       2:                  10 ~ {2}
       3:                  11 ~ {1,2}
       4:                 100 ~ {3}
       5:                 101 ~ {1,3}
       8:                1000 ~ {4}
       9:                1001 ~ {1,4}
      10:                1010 ~ {2,4}
      11:                1011 ~ {1,2,4}
      16:               10000 ~ {5}
      17:               10001 ~ {1,5}
      64:             1000000 ~ {7}
      65:             1000001 ~ {1,7}
     128:            10000000 ~ {8}
     129:            10000001 ~ {1,8}
     130:            10000010 ~ {2,8}
     131:            10000011 ~ {1,2,8}
     136:            10001000 ~ {4,8}
     137:            10001001 ~ {1,4,8}
     138:            10001010 ~ {2,4,8}
     139:            10001011 ~ {1,2,4,8}
     256:           100000000 ~ {9}
     257:           100000001 ~ {1,9}
     260:           100000100 ~ {3,9}
     261:           100000101 ~ {1,3,9}
    1024:         10000000000 ~ {11}
    1025:         10000000001 ~ {1,11}
    4096:       1000000000000 ~ {13}
    4097:       1000000000001 ~ {1,13}
   32768:    1000000000000000 ~ {16}

For powers of 2 we have A253317.
For prime indices we have A320698.
For squarefree numbers instead of prime powers we have A371289.
A000040 lists prime numbers.
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. A005117, A326782, A368533, A371292, A371443, A371452.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
Select[Range[1000],#==1||PrimePowerQ[Times@@bpe[#]]&]

cp's OEIS Frontend

A371452 Number of connected components of the prime indices of the binary indices of n.

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

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A371289 Numbers whose binary indices have squarefree product.

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

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A371293 Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.

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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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A371290 Numbers whose product of binary indices is a prime power > 1.

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