A277098 -id:A277098 - cp's OEIS frontend

A324697 Lexicographically earliest sequence of positive integers > 1 that are prime or whose prime indices already belong to the sequence.

2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 27, 29, 31, 33, 37, 41, 43, 45, 47, 51, 53, 55, 59, 61, 67, 69, 71, 73, 75, 79, 81, 83, 85, 89, 93, 97, 99, 101, 103, 107, 109, 113, 115, 121, 123, 125, 127, 131, 135, 137, 139, 141, 149, 151, 153, 155, 157, 163, 165

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:
   2: {1}
   3: {2}
   5: {3}
   7: {4}
   9: {2,2}
  11: {5}
  13: {6}
  15: {2,3}
  17: {7}
  19: {8}
  23: {9}
  25: {3,3}
  27: {2,2,2}
  29: {10}
  31: {11}
  33: {2,5}
  37: {12}
  41: {13}
  43: {14}
  45: {2,2,3}

Complement of A324705.
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,False,?PrimeQ,True,,And@@Cases[FactorInteger[n],{p_,k_}:>aQ[PrimePi[p]]]];
Select[Range[100],aQ]

A306202 Matula-Goebel numbers of rooted semi-identity trees.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 22, 24, 26, 28, 29, 30, 31, 32, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 47, 48, 51, 52, 53, 55, 56, 57, 58, 59, 60, 62, 64, 65, 66, 67, 68, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 84, 85

Offset: 1

Gus Wiseman, Jan 29 2019

nonn

Definition: A positive integer belongs to the sequence iff its prime indices greater than 1 are distinct and already belong to the sequence. 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 all unlabeled rooted semi-identity trees together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((o)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
  10: (o((o)))
  11: ((((o))))
  12: (oo(o))
  13: ((o(o)))
  14: (o(oo))
  15: ((o)((o)))
  16: (oooo)
  17: (((oo)))
  19: ((ooo))
  20: (oo((o)))
  21: ((o)(oo))
  22: (o(((o))))
  24: (ooo(o))
  26: (o(o(o)))
  28: (oo(oo))
  29: ((o((o))))
  30: (o(o)((o)))

Index entries for sequences related to Matula-Goebel numbers

Cf. A000081, A004111, A007097, A276625, A277098, A306200, A306201, A316467.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
psidQ[n_]:=And[UnsameQ@@DeleteCases[primeMS[n],1],And@@psidQ/@primeMS[n]];
Select[Range[100],psidQ]

A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 196, 256, 343, 361, 392, 512, 784, 1024, 1372, 1444, 1568, 2048, 2401, 2744, 2809, 2888, 3136, 4096, 5488, 5776, 6272, 6859, 8192, 9604, 10976, 11236, 11552, 12544, 16384, 16807, 17161, 17689, 19208, 21952, 22472, 23104, 25088

Offset: 1

Gus Wiseman, Aug 04 2018

nonn

A positive integer n is a Matula-Goebel number of a series-reduced powerful rooted tree iff either n = 1 or n is a powerful number (meaning its prime multiplicities are all greater than 1) whose prime indices are all Matula-Goebel numbers of series-reduced powerful rooted trees, where a prime index of n is a number m such that prime(m) divides n.

			The sequence of Matula-Goebel numbers of series-reduced powerful rooted trees together with the corresponding trees begins:
    1: o
    4: (oo)
    8: (ooo)
   16: (oooo)
   32: (ooooo)
   49: ((oo)(oo))
   64: (oooooo)
  128: (ooooooo)
  196: (oo(oo)(oo))
  256: (oooooooo)
  343: ((oo)(oo)(oo))
  361: ((ooo)(ooo))
  392: (ooo(oo)(oo))
  512: (ooooooooo)
  784: (oooo(oo)(oo))

Index entries for sequences related to Matula-Göbel numbers

Cf. A000081, A001694, A061775, A111299, A214577, A276625, A277098, A303431.

Cf. A317102, A317707, A317708, A317709, A317710, A317711, A317712, A317717, A317718, A317719.

powgoQ[n_]:=Or[n==1,And[Min@@FactorInteger[n][[All,2]]>1,And@@powgoQ/@PrimePi/@FactorInteger[n][[All,1]]]];
Select[Range[1000],powgoQ] (* Gus Wiseman, Aug 31 2018 *)
(* Second program: *)
Nest[Function[a, Append[a, Block[{k = a[[-1]] + 1}, While[Nand[AllTrue[#[[All, -1]], # > 1 & ], AllTrue[PrimePi[#[[All, 1]] ], MemberQ[a, #] &]] &@ FactorInteger@ k, k++]; k]]], {1}, 44] (* Michael De Vlieger, Aug 05 2018 *)

Rewritten by Gus Wiseman, Aug 31 2018

A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 4, 1, 0, 0, 0, 1, 6, 5, 0, 0, 0, 0, 1, 9, 13, 2, 0, 0, 0, 0, 1, 12, 28, 11, 0, 0, 0, 0, 0, 1, 16, 53, 40, 3, 0, 0, 0, 0, 0, 1, 20, 91, 109, 26, 0, 0, 0, 0, 0, 0, 1, 25, 146, 254, 116, 6, 0, 0, 0, 0, 0, 0, 1, 30, 223, 524, 387, 61, 0, 0, 0, 0, 0, 0, 0, 1, 36

Offset: 1

Gus Wiseman, Mar 19 2018

			Triangle begins:
1
1   0
1   0   0
1   1   0   0
1   2   0   0   0
1   4   1   0   0   0
1   6   5   0   0   0   0
1   9  13   2   0   0   0   0
1  12  28  11   0   0   0   0   0
1  16  53  40   3   0   0   0   0   0
1  20  91 109  26   0   0   0   0   0   0
1  25 146 254 116   6   0   0   0   0   0   0
1  30 223 524 387  61   0   0   0   0   0   0   0
The T(6,2) = 4 rooted identity trees: (((o(o)))), ((o((o)))), (o(((o)))), ((o)((o))).

A version with the zeroes removed is A055327.

Cf. A000081, A001190, A003238, A004111, A032305, A055277, A273873, A276625, A277098, A290689, A298118, A298422, A298426, A301343, A301344, A301345.

irut[n_]:=irut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[irut/@c]],UnsameQ@@#&]]/@IntegerPartitions[n-1]];
Table[Length[Select[irut[n],Count[#,{},{-2}]===k&]],{n,8},{k,n}]

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]

cp's OEIS Frontend

A324697 Lexicographically earliest sequence of positive integers > 1 that are prime or whose prime indices already belong to the sequence.

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A306202 Matula-Goebel numbers of rooted semi-identity trees.

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A317705 Matula-Goebel numbers of series-reduced powerful rooted trees.

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A301342 Regular triangle where T(n,k) is the number of rooted identity trees with n nodes and k leaves.

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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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