A324694 -id:A324694 - cp's OEIS frontend

A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

1, 2, 3, 5, 8, 13, 26, 42, 72, 120, 232, 376, 752, 1128, 2256, 4512, 8256, 13632, 27264, 42048, 82944, 158976, 313344, 497664, 995328, 1700352, 3350016, 5815296, 11630592, 17491968, 34983936, 56954880, 108933120, 210788352, 418258944, 804667392, 1609334784

Offset: 0

Gus Wiseman, Mar 13 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(0) = 1 through a(6) = 26 subsets:
  {}  {}   {}   {}     {}     {}       {}
      {1}  {1}  {1}    {1}    {1}      {1}
           {2}  {2}    {2}    {2}      {2}
                {3}    {3}    {3}      {3}
                {1,3}  {4}    {4}      {4}
                       {1,3}  {5}      {5}
                       {2,4}  {1,3}    {6}
                       {3,4}  {1,5}    {1,3}
                              {2,4}    {1,5}
                              {2,5}    {1,6}
                              {3,4}    {2,4}
                              {4,5}    {2,5}
                              {2,4,5}  {2,6}
                                       {3,4}
                                       {3,6}
                                       {4,5}
                                       {4,6}
                                       {5,6}
                                       {1,3,6}
                                       {1,5,6}
                                       {2,4,5}
                                       {2,4,6}
                                       {2,5,6}
                                       {3,4,6}
                                       {4,5,6}
                                       {2,4,5,6}

Andrew Howroyd, Table of n, a(n) for n = 0..100

The maximal case is A324744. The case of subsets of {2...n} is A324739. The strict integer partition version is A324749. The integer partition version is A324754. The Heinz number version is A324759. An infinite version is A324694.
Cf. A000720, A001221, A001462, A007097, A076078, A084422, A085945, A112798, A276625, A279861, A290689, A290822, A304360, A306844.
Cf. A324695, A324736, A324741, A324750, A324755, A324760.

Table[Length[Select[Subsets[Range[n]],!MemberQ[#,k_/;SubsetQ[#,PrimePi/@First/@FactorInteger[k]]]&]],{n,0,10}]

pset(n)={my(b=0,f=factor(n)[,1]); sum(i=1, #f, 1<<(primepi(f[i])))}
a(n)={my(p=vector(n,k,if(k==1, 1, pset(k))), d=0); for(i=1, #p, d=bitor(d, p[i]));
((k,b)->if(k>#p, 1, my(t=self()(k+1,b)); if(bitnegimply(p[k], b), t+=if(bittest(d,k), self()(k+1, b+(1<Andrew Howroyd, Aug 16 2019

Terms a(21) and beyond from Andrew Howroyd, Aug 16 2019

A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

Original entry on oeis.org

1, 0, 1, 2, 5, 10, 28, 64, 169, 422, 1108, 2872, 7627, 20202, 54216, 145867, 395288

Offset: 1

Gus Wiseman, Mar 18 2019

			The a(1) = 1 through a(6) = 10 rooted trees:
  o  ((o))  ((oo))   ((ooo))    ((oooo))
            (((o)))  (((oo)))   (((ooo)))
                     ((o)(o))   ((o)(oo))
                     ((o(o)))   ((o(oo)))
                     ((((o))))  ((oo(o)))
                                ((((oo))))
                                (((o)(o)))
                                (((o(o))))
                                ((o((o))))
                                (((((o)))))

Cf. A000081, A290689, A304360, A306844, A317787, A318185.

Cf. A324694, A324696, A324704, A324738, A324744, A324758, A324759, A324765, A324768, A324771, A324839, A324840, A324844, A324846.

submultQ[cap_,fat_]:=And@@Function[i,Count[fat,i]>=Count[cap,i]]/@Union[List@@cap];
rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],And@@Table[!submultQ[b,#],{b,#}]&]],{n,10}]

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

Original entry on oeis.org

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]

cp's OEIS Frontend

A324738 Number of subsets of {1...n} containing no element > 1 whose prime indices all belong to the subset.

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A324838 Number of unlabeled rooted trees with n nodes where the branches of no branch of the root form a submultiset of the branches of the root.

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

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

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

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

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

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

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