A324695 -id:A324695 - cp's OEIS frontend

A331912 Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.

1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 26, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 52, 53, 58, 59, 61, 64, 65, 67, 71, 73, 74, 79, 81, 83, 86, 87, 89, 91, 94, 97, 101, 103, 104, 107, 109, 111, 113, 116, 117, 121, 122, 125, 127, 128, 129, 131, 137

Offset: 1

Gus Wiseman, Feb 01 2020

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.

Conjecture: a(n)/A331784(n) -> 1 as n -> infinity.

			The sequence of terms together with their prime indices begins:
    1: {}              37: {12}              86: {1,14}
    2: {1}             39: {2,6}             87: {2,10}
    3: {2}             41: {13}              89: {24}
    4: {1,1}           43: {14}              91: {4,6}
    5: {3}             47: {15}              94: {1,15}
    7: {4}             49: {4,4}             97: {25}
    8: {1,1,1}         52: {1,1,6}          101: {26}
    9: {2,2}           53: {16}             103: {27}
   11: {5}             58: {1,10}           104: {1,1,1,6}
   13: {6}             59: {17}             107: {28}
   16: {1,1,1,1}       61: {18}             109: {29}
   17: {7}             64: {1,1,1,1,1,1}    111: {2,12}
   19: {8}             65: {3,6}            113: {30}
   23: {9}             67: {19}             116: {1,1,10}
   25: {3,3}           71: {20}             117: {2,2,6}
   26: {1,6}           73: {21}             121: {5,5}
   27: {2,2,2}         74: {1,12}           122: {1,18}
   29: {10}            79: {22}             125: {3,3,3}
   31: {11}            81: {2,2,2,2}        127: {31}
   32: {1,1,1,1,1}     83: {23}             128: {1,1,1,1,1,1,1}
For example, the prime indices of 117 are {2,2,6}, of which only 2 is already in the sequence, so 117 is in the sequence.

Gus Wiseman, Plot of A331912(n)/A331784(n) for n = 1..3729.

Contains all prime powers A000961.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with at most one prime index in S are A331784.
Numbers S with exactly one prime index in S are A331785.
Numbers S with exactly one distinct prime index in S are A331913.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.

aQ[n_]:=Length[Select[PrimePi/@First/@If[n==1,{},FactorInteger[n]],aQ]]<=1;
Select[Range[100],aQ]

A324699 Lexicographically earliest sequence of positive integers whose prime indices minus 1 already belong to the sequence.

Original entry on oeis.org

1, 3, 7, 9, 19, 21, 27, 29, 49, 57, 63, 71, 79, 81, 87, 107, 113, 133, 147, 171, 189, 203, 213, 229, 237, 243, 261, 271, 311, 321, 339, 343, 359, 361, 399, 409, 421, 441, 457, 497, 513, 551, 553, 567, 593, 609, 619, 639, 687, 711, 729, 749, 757, 783, 791, 813

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

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 sequence of terms together with their prime indices begins:
    1: {}
    3: {2}
    7: {4}
    9: {2,2}
   19: {8}
   21: {2,4}
   27: {2,2,2}
   29: {10}
   49: {4,4}
   57: {2,8}
   63: {2,2,4}
   71: {20}
   79: {22}
   81: {2,2,2,2}
   87: {2,10}
  107: {28}
  113: {30}
  133: {4,8}
  147: {2,4,4}
  171: {2,2,8}
  189: {2,2,2,4}

Prime indices are A306719.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360, A306719.
Cf. A324694, A324695, A324696, A324697, A324698, A324700, A324701, A324702, A324703, A324704, A324705.

aQ[n_]:=Switch[n,0,False,1,True,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]-1]]];
Select[Range[100],aQ]

a(n) = A306719(n) - 1.

A324700 Lexicographically earliest sequence containing 0 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

Original entry on oeis.org

0, 2, 4, 5, 8, 10, 11, 13, 16, 20, 22, 23, 25, 26, 31, 32, 37, 40, 43, 44, 46, 50, 52, 55, 59, 62, 64, 65, 73, 74, 80, 83, 86, 88, 89, 92, 100, 101, 103, 104, 110, 115, 118, 121, 124, 125, 128, 130, 131, 137, 143, 146, 148, 155, 160, 163, 166, 169, 172, 176

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

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 sequence of terms together with their prime indices begins:
   0
   2: {1}
   4: {1,1}
   5: {3}
   8: {1,1,1}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  20: {1,1,3}
  22: {1,5}
  23: {9}
  25: {3,3}
  26: {1,6}
  31: {11}
  32: {1,1,1,1,1}
  37: {12}
  40: {1,1,1,3}
  43: {14}
  44: {1,1,5}

Prime indices are A324701.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324702, A324703, A324704, A324705.

aQ[n_]:=Switch[n,0,True,1,False,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]-1]]];
Select[Range[0,100],aQ]

a(n) = A324701(n) - 1.

A324701 Lexicographically earliest sequence containing 1 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Original entry on oeis.org

1, 3, 5, 6, 9, 11, 12, 14, 17, 21, 23, 24, 26, 27, 32, 33, 38, 41, 44, 45, 47, 51, 53, 56, 60, 63, 65, 66, 74, 75, 81, 84, 87, 89, 90, 93, 101, 102, 104, 105, 111, 116, 119, 122, 125, 126, 129, 131, 132, 138, 144, 147, 149, 156, 161, 164, 167, 170, 173, 177

Offset: 1

Gus Wiseman, Mar 10 2019

nonn

A self-describing sequence, similar to A304360.

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.

Prime indices of A324700.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324702, A324703, A324704, A324705.

aQ[n_]:=Switch[n,0,False,1,True,,And@@Cases[FactorInteger[n-1],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[0,100],aQ]

a(n) = A324700(n) + 1.

A324702 Lexicographically earliest sequence containing 2 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

Original entry on oeis.org

2, 5, 13, 25, 43, 65, 101, 125, 169, 193, 215, 317, 325, 505, 557, 559, 625, 701, 845, 965, 1013, 1075, 1181, 1313, 1321, 1585, 1625, 1849, 2111, 2161, 2197, 2509, 2525, 2785, 2795, 3125, 3505, 3617, 4049, 4057, 4121, 4225, 4343, 4639, 4825, 5065, 5297, 5375

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.
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.
Also 2 and numbers whose prime indices belong to A324703.

			The sequence of terms together with their prime indices begins:
    2: {1}
    5: {3}
   13: {6}
   25: {3,3}
   43: {14}
   65: {3,6}
  101: {26}
  125: {3,3,3}
  169: {6,6}
  193: {44}
  215: {3,14}
  317: {66}
  325: {3,3,6}
  505: {3,26}
  557: {102}
  559: {6,14}
  625: {3,3,3,3}
  701: {126}
  845: {3,6,6}
  965: {3,44}

Prime indices > 1 are A324703.
Cf. A000002, A000720, A001222, A001462, A007097, A045965, A055396, A061395, A064989, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324700, A324701, A324704, A324705.

aQ[n_]:=Switch[n,0,False,1,False,2,True,,And@@Cases[FactorInteger[n],{p,k_}:>aQ[PrimePi[p]-1]]];
Select[Range[100],aQ]

a(n) = A324703(n) - 1.

A324703 Lexicographically earliest sequence containing 3 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Original entry on oeis.org

3, 6, 14, 26, 44, 66, 102, 126, 170, 194, 216, 318, 326, 506, 558, 560, 626, 702, 846, 966, 1014, 1076, 1182, 1314, 1322, 1586, 1626, 1850, 2112, 2162, 2198, 2510, 2526, 2786, 2796, 3126, 3506, 3618, 4050, 4058, 4122, 4226, 4344, 4640, 4826, 5066, 5298, 5376

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.

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.

Prime indices of A324702.
Cf. A000002, A000720, A001222, A001462, A007097, A045965, A055396, A061395, A064989, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324697, A324698, A324699, A324700, A324701, A324704, A324705.

aQ[n_]:=Switch[n,0,False,3,True,,And@@Cases[FactorInteger[n-1],{p,k_}:>aQ[PrimePi[p]]]];
Select[Range[0,1000],aQ]

a(n) = A324702(n) + 1.

A324705 Lexicographically earliest sequence containing 1 and all composite numbers divisible by prime(m) for some m already in the sequence.

Original entry on oeis.org

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 21, 22, 24, 26, 28, 30, 32, 34, 35, 36, 38, 39, 40, 42, 44, 46, 48, 49, 50, 52, 54, 56, 57, 58, 60, 62, 63, 64, 65, 66, 68, 70, 72, 74, 76, 77, 78, 80, 82, 84, 86, 87, 88, 90, 91, 92, 94, 95, 96, 98, 100, 102, 104, 105, 106

Offset: 1

Gus Wiseman, Mar 11 2019

nonn

A self-describing sequence, similar to A304360.

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 sequence of terms together with their prime indices begins:
   1: {}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  10: {1,3}
  12: {1,1,2}
  14: {1,4}
  16: {1,1,1,1}
  18: {1,2,2}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  24: {1,1,1,2}
  26: {1,6}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  34: {1,7}
  35: {3,4}
  36: {1,1,2,2}

Complement of A324697.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098, A304360.
Cf. A324694, A324695, A324696, A324698, A324699, A324700, A324701, A324702, A324703, A324704.

aQ[n_]:=Switch[n,1,True,?PrimeQ,False,,!And@@Cases[FactorInteger[n],{p_,k_}:>!aQ[PrimePi[p]]]];
Select[Range[200],aQ]

A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 111, 113, 115, 119, 122, 127, 131, 133, 137, 139, 141, 142

Offset: 1

Gus Wiseman, Feb 01 2020

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.

Conjecture: A331912(n)/a(n) -> 1 as n -> infinity.

			The sequence of terms together with their prime indices begins:
    1: {}        43: {14}       91: {4,6}      141: {2,15}
    2: {1}       46: {1,9}      94: {1,15}     142: {1,20}
    3: {2}       47: {15}       95: {3,8}      143: {5,6}
    5: {3}       49: {4,4}      97: {25}       145: {3,10}
    7: {4}       53: {16}       98: {1,4,4}    147: {2,4,4}
   11: {5}       57: {2,8}     101: {26}       149: {35}
   13: {6}       58: {1,10}    103: {27}       151: {36}
   14: {1,4}     59: {17}      106: {1,16}     157: {37}
   17: {7}       61: {18}      107: {28}       158: {1,22}
   19: {8}       65: {3,6}     109: {29}       159: {2,16}
   21: {2,4}     67: {19}      111: {2,12}     161: {4,9}
   23: {9}       69: {2,9}     113: {30}       163: {38}
   26: {1,6}     71: {20}      115: {3,9}      167: {39}
   29: {10}      73: {21}      119: {4,7}      169: {6,6}
   31: {11}      74: {1,12}    122: {1,18}     173: {40}
   35: {3,4}     77: {4,5}     127: {31}       178: {1,24}
   37: {12}      79: {22}      131: {32}       179: {41}
   38: {1,8}     83: {23}      133: {4,8}      181: {42}
   39: {2,6}     87: {2,10}    137: {33}       182: {1,4,6}
   41: {13}      89: {24}      139: {34}       183: {2,18}
For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.

Gus Wiseman, Plot of A331912(n)/A331784(n) for n = 1..3729.

Contains all prime numbers A000040.
Numbers S without all prime indices in S are A324694.
Numbers S without any prime indices in S are A324695.
Numbers S with exactly one prime index in S are A331785.
Numbers S with at most one distinct prime index in S are A331912.
Numbers S with exactly one distinct prime index in S are A331913.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331914.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
aQ[n_]:=Length[Cases[primeMS[n],_?aQ]]<=1;
Select[Range[100],aQ]

A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 6, 6, 6, 6, 10, 10, 16, 16, 16, 16, 24, 24, 48, 48, 48, 48, 84, 84, 84, 84, 84, 84, 144, 144, 228, 228, 228, 228, 228, 228, 420, 420, 420, 420, 648, 648, 1080, 1080, 1080, 1080, 1800, 1800, 1800, 1800, 1800, 1800, 3600, 3600, 3600, 3600, 3600

Offset: 1

Gus Wiseman, Mar 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 a(1) = 1 through a(9) = 6 subsets:
  {}  {2}  {2}  {2,4}  {3,4}    {2,4,5}  {2,4,5}  {2,4,5,8}  {2,4,5,8}
           {3}  {3,4}  {2,4,5}  {3,4,6}  {2,5,7}  {2,5,7,8}  {2,5,7,8}
                                {4,5,6}  {3,4,6}  {3,4,6,8}  {3,4,6,8,9}
                                         {3,6,7}  {3,6,7,8}  {3,6,7,8,9}
                                         {4,5,6}  {4,5,6,8}  {4,5,6,8,9}
                                         {5,6,7}  {5,6,7,8}  {5,6,7,8,9}

Andrew Howroyd, Table of n, a(n) for n = 1..100

The non-maximal version is A324742.
The version for subsets of {1...n} is A324741.
An infinite version is A304360.
Cf. A076078, A084422, A085945, A112798, A276625, A306844, A324764.
Cf. A324695, A324743, A324751, A324756, A324758, A324762.

maxim[s_]:=Complement[s,Last/@Select[Tuples[s,2],UnsameQ@@#&&SubsetQ@@#&]];
Table[Length[maxim[Select[Subsets[Range[2,n]],Intersection[#,PrimePi/@First/@Join@@FactorInteger/@#]=={}&]]],{n,10}]

pset(n)={my(b=0, f=factor(n)[, 1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n-1, k, pset(k+1)>>1), d=0); for(i=1, #p, d=bitor(d, p[i]));
my(ismax(b)=my(e=0); forstep(k=#p, 1, -1, if(bittest(b,k), e=bitor(e,p[k]), if(!bittest(e,k) && !bitand(p[k], b), return(0)) )); 1);
((k, b)->if(k>#p, ismax(b), my(f=!bitand(p[k], b)); if(!f || bittest(d, k), self()(k+1, b)) + if(f, self()(k+1, b+(1<Andrew Howroyd, Aug 26 2019

Terms a(16) and beyond from Andrew Howroyd, Aug 26 2019

A358453 Number of transitive ordered rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 8, 17, 37, 83, 190, 444, 1051, 2518, 6090, 14852

Offset: 1

Gus Wiseman, Nov 18 2022

We define an unlabeled ordered rooted tree to be transitive if every branch of a branch of the root already appears farther to the left as a branch of the root. An undirected version is A358454.

			The a(1) = 1 through a(7) = 17 trees:
  o  (o)  (oo)  (ooo)   (oooo)   (ooooo)    (oooooo)
                (o(o))  (o(o)o)  (o(o)oo)   (o(o)ooo)
                        (o(oo))  (o(oo)o)   (o(oo)oo)
                        (oo(o))  (o(ooo))   (o(ooo)o)
                                 (oo(o)o)   (o(oooo))
                                 (oo(oo))   (oo(o)oo)
                                 (ooo(o))   (oo(oo)o)
                                 (o(o)(o))  (oo(ooo))
                                            (ooo(o)o)
                                            (ooo(oo))
                                            (oooo(o))
                                            (o(o)(o)o)
                                            (o(o)(oo))
                                            (o(o)o(o))
                                            (o(oo)(o))
                                            (oo(o)(o))
                                            (o(o)((o)))

The unordered version is A290689, ranked by A290822.
The undirected version is A358454, ranked by A358458.
These trees are ranked by A358457.
A000081 counts rooted trees.
A306844 counts anti-transitive rooted trees.
Cf. A318185, A324695, A324751, A324756, A324758, A324764-A324768, A324838, A324840, A324844, A358456.

aot[n_]:=If[n==1,{{}},Join@@Table[Tuples[aot/@c],{c,Join@@Permutations/@IntegerPartitions[n-1]}]];
Table[Length[Select[aot[n],Function[t,And@@Table[Complement[t[[k]],Take[t,k]]=={},{k,Length[t]}]]]],{n,10}]

cp's OEIS Frontend

A331912 Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A324699 Lexicographically earliest sequence of positive integers whose prime indices minus 1 already belong to the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A324700 Lexicographically earliest sequence containing 0 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A324701 Lexicographically earliest sequence containing 1 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A324702 Lexicographically earliest sequence containing 2 and all positive integers > 1 whose prime indices minus 1 already belong to the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A324703 Lexicographically earliest sequence containing 3 and all positive integers n such that the prime indices of n - 1 already belong to the sequence.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A324705 Lexicographically earliest sequence containing 1 and all composite numbers divisible by prime(m) for some m already in the sequence.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A331784 Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A324763 Number of maximal subsets of {2...n} containing no prime indices of the elements.

Views