A371451 -id:A371451 - 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}]

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[#]]&]

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.

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[#]]&]

A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99

Offset: 1

Gus Wiseman, Apr 01 2024

nonn

In an antichain of sets, no edge is a proper subset of any other.

			The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Jakub Buczak, Table of n, a(n) for n = 1..10000

Contains all powers of primes A000961.
An opposite version is A087086, carry-connected case A371294.
For prime indices of prime indices we have A316476, carry-connected A329559.
These antichains are counted by A325109.
For binary indices of binary indices we have A326704, carry-conn. A326750.
The carry-connected case is A371445, counted by A371446.
A048143 counts connected antichains of sets.
A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
A050320 counts set multipartitions of prime indices, see also A318360.
A070939 gives length of binary expansion.
A089259 counts set multipartitions of integer partitions.
A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
A116540 counts normal set multipartitions.
A302478 ranks set multipartitions, cf. A073576.
A325118 ranks carry-connected partitions, counted by A325098.
A371451 counts carry-connected components of binary indices.
Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],stableQ[bix/@prix[#],SubsetQ]&]

cp's OEIS Frontend

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

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

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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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A371449 Numbers whose prime indices are not powers of 2.

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A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

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