cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A331915 Numbers with exactly one prime prime index, counted with multiplicity.

Original entry on oeis.org

3, 5, 6, 10, 11, 12, 17, 20, 21, 22, 24, 31, 34, 35, 39, 40, 41, 42, 44, 48, 57, 59, 62, 65, 67, 68, 69, 70, 77, 78, 80, 82, 83, 84, 87, 88, 95, 96, 109, 111, 114, 115, 118, 119, 124, 127, 129, 130, 134, 136, 138, 140, 141, 143, 145, 147, 154, 156, 157, 159
Offset: 1

Views

Author

Gus Wiseman, Feb 08 2020

Keywords

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}             57: {2,8}            114: {1,2,8}
    5: {3}             59: {17}             115: {3,9}
    6: {1,2}           62: {1,11}           118: {1,17}
   10: {1,3}           65: {3,6}            119: {4,7}
   11: {5}             67: {19}             124: {1,1,11}
   12: {1,1,2}         68: {1,1,7}          127: {31}
   17: {7}             69: {2,9}            129: {2,14}
   20: {1,1,3}         70: {1,3,4}          130: {1,3,6}
   21: {2,4}           77: {4,5}            134: {1,19}
   22: {1,5}           78: {1,2,6}          136: {1,1,1,7}
   24: {1,1,1,2}       80: {1,1,1,1,3}      138: {1,2,9}
   31: {11}            82: {1,13}           140: {1,1,3,4}
   34: {1,7}           83: {23}             141: {2,15}
   35: {3,4}           84: {1,1,2,4}        143: {5,6}
   39: {2,6}           87: {2,10}           145: {3,10}
   40: {1,1,1,3}       88: {1,1,1,5}        147: {2,4,4}
   41: {13}            95: {3,8}            154: {1,4,5}
   42: {1,2,4}         96: {1,1,1,1,1,2}    156: {1,1,2,6}
   44: {1,1,5}        109: {29}             157: {37}
   48: {1,1,1,1,2}    111: {2,12}           159: {2,16}

Crossrefs

These are numbers n such that A257994(n) = 1.
Prime-indexed primes are A006450, with products A076610.
The number of distinct prime prime indices is A279952.
Numbers with at least one prime prime index are A331386.
The set S of numbers with exactly one prime index in S are A331785.
The set S of numbers with exactly one distinct prime index in S are A331913.
Numbers with at most one prime prime index are A331914.
Numbers with exactly one distinct prime prime index are A331916.
Numbers with at most one distinct prime prime index are A331995.
Cf. A000040, A000720, A007097, A007821, A018252, A112798, A289509, A320628, A330944, A330945, A331784.

Programs

  • Mathematica
    primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Count[primeMS[#],_?PrimeQ]==1&]

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

Views

Author

Gus Wiseman, Feb 01 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

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

A331936 Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex (semi-achirality).

Original entry on oeis.org

1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 144, 148, 152, 162, 169, 172, 178, 184, 192, 196, 202, 206, 208, 212, 214, 216, 224, 243, 244, 256, 262, 288
Offset: 1

Views

Author

Gus Wiseman, Feb 03 2020

Keywords

Comments

First differs from A331873 in lacking 69, the Matula-Goebel number of the tree ((o)((o)(o))).
A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of 1, 2, and all numbers equal to a power of 2 (other than 1) times a power of prime(j) for some j > 1 already in the sequence.

Examples

			The sequence of rooted trees ranked by this sequence together with their Matula-Goebel numbers begins:
   1: o
   2: (o)
   4: (oo)
   6: (o(o))
   8: (ooo)
   9: ((o)(o))
  12: (oo(o))
  14: (o(oo))
  16: (oooo)
  18: (o(o)(o))
  24: (ooo(o))
  26: (o(o(o)))
  27: ((o)(o)(o))
  28: (oo(oo))
  32: (ooooo)
  36: (oo(o)(o))
  38: (o(ooo))
  46: (o((o)(o)))
  48: (oooo(o))
  49: ((oo)(oo))
The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}            152: {1,1,1,8}
    2: {1}             54: {1,2,2,2}          162: {1,2,2,2,2}
    4: {1,1}           56: {1,1,1,4}          169: {6,6}
    6: {1,2}           64: {1,1,1,1,1,1}      172: {1,1,14}
    8: {1,1,1}         72: {1,1,1,2,2}        178: {1,24}
    9: {2,2}           74: {1,12}             184: {1,1,1,9}
   12: {1,1,2}         76: {1,1,8}            192: {1,1,1,1,1,1,2}
   14: {1,4}           81: {2,2,2,2}          196: {1,1,4,4}
   16: {1,1,1,1}       86: {1,14}             202: {1,26}
   18: {1,2,2}         92: {1,1,9}            206: {1,27}
   24: {1,1,1,2}       96: {1,1,1,1,1,2}      208: {1,1,1,1,6}
   26: {1,6}           98: {1,4,4}            212: {1,1,16}
   27: {2,2,2}        104: {1,1,1,6}          214: {1,28}
   28: {1,1,4}        106: {1,16}             216: {1,1,1,2,2,2}
   32: {1,1,1,1,1}    108: {1,1,2,2,2}        224: {1,1,1,1,1,4}
   36: {1,1,2,2}      112: {1,1,1,1,4}        243: {2,2,2,2,2}
   38: {1,8}          122: {1,18}             244: {1,1,18}
   46: {1,9}          128: {1,1,1,1,1,1,1}    256: {1,1,1,1,1,1,1,1}
   48: {1,1,1,1,2}    144: {1,1,1,1,2,2}      262: {1,32}
   49: {4,4}          148: {1,1,12}           288: {1,1,1,1,1,2,2}

Links

Crossrefs

A superset of A000079.
The non-lone-child-avoiding version is A320230.
The non-semi version is A320269.
These trees are counted by A331933.
Not requiring semi-achirality gives A331935.
The fully-achiral case is A331992.
Achiral trees are counted by A003238.
Numbers with at most one distinct odd prime factor are A070776.
Matula-Goebel numbers of achiral rooted trees are A214577.
Matula-Goebel numbers of semi-identity trees are A306202.
Numbers S with at most one distinct prime index in S are A331912.
Cf. A001678, A007097, A050381, A061775, A196050, A291636, A331784, A331873, A331914, A331934, A331965, A331967, A331991.

Programs

  • Mathematica
    msQ[n_]:=n<=2||!PrimeQ[n]&&Length[DeleteCases[FactorInteger[n],{2,_}]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100],msQ]

Formula

Intersection of A320230 and A331935.

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

Views

Author

Gus Wiseman, Feb 01 2020

Keywords

Comments

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.

Examples

			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.

Links

Crossrefs

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.

Programs

  • Mathematica
    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]

A331933 Number of semi-lone-child-avoiding rooted trees with at most one distinct non-leaf branch directly under any vertex.

Original entry on oeis.org

1, 1, 1, 2, 4, 6, 12, 18, 33, 52, 90, 142, 242, 384, 639, 1028, 1688, 2716, 4445, 7161, 11665, 18839, 30595, 49434, 80199, 129637, 210079, 339750, 550228, 889978, 1440909, 2330887, 3772845, 6103823, 9878357, 15982196, 25863454, 41845650, 67713550, 109559443
Offset: 1

Views

Author

Gus Wiseman, Feb 03 2020

Keywords

Comments

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.

Examples

			The a(1) = 1 through a(8) = 18 trees:
  o  (o)  (oo)  (ooo)   (oooo)    (ooooo)    (oooooo)
                (o(o))  (o(oo))   (o(ooo))   (o(oooo))
                        (oo(o))   (oo(oo))   (oo(ooo))
                        ((o)(o))  (ooo(o))   (ooo(oo))
                                  (o(o)(o))  (oooo(o))
                                  (o(o(o)))  ((oo)(oo))
                                             (o(o(oo)))
                                             (o(oo(o)))
                                             (oo(o)(o))
                                             (oo(o(o)))
                                             ((o)(o)(o))
                                             (o((o)(o)))

Links

Crossrefs

Not requiring lone-child-avoidance gives A320222.
The non-semi version is A320268.
Matula-Goebel numbers of these trees are A331936.
Achiral trees are A003238.
Semi-identity trees are A306200.
Numbers S with at most one distinct prime index in S are A331912.
Semi-lone-child-avoiding rooted trees are A331934.
Cf. A000081, A001678, A004111, A050381, A214577, A291636, A320230, A320269, A331784, A331872, A331914, A331935, A331966.

Programs

  • Mathematica
    sseo[n_]:=Switch[n,1,{{}},2,{{{}}},_,Join@@Function[c,Select[Union[Sort/@Tuples[sseo/@c]],Length[Union[DeleteCases[#,{}]]]<=1&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[sseo[n]],{n,10}]
  • PARI
    seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + sum(i=2, n-2, ((n-1)\i)*v[i])); v} \\ Andrew Howroyd, Feb 09 2020

Formula

a(n) = 1 + Sum_{i=2..n-2} floor((n-1)/i)*a(i). - Andrew Howroyd, Feb 09 2020

Extensions

Terms a(31) and beyond from Andrew Howroyd, Feb 09 2020

A331916 Numbers with exactly one distinct prime prime index.

Original entry on oeis.org

3, 5, 6, 9, 10, 11, 12, 17, 18, 20, 21, 22, 24, 25, 27, 31, 34, 35, 36, 39, 40, 41, 42, 44, 48, 50, 54, 57, 59, 62, 63, 65, 67, 68, 69, 70, 72, 77, 78, 80, 81, 82, 83, 84, 87, 88, 95, 96, 100, 108, 109, 111, 114, 115, 117, 118, 119, 121, 124, 125, 126, 127
Offset: 1

Views

Author

Gus Wiseman, Feb 08 2020

Keywords

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
    3: {2}           40: {1,1,1,3}       81: {2,2,2,2}
    5: {3}           41: {13}            82: {1,13}
    6: {1,2}         42: {1,2,4}         83: {23}
    9: {2,2}         44: {1,1,5}         84: {1,1,2,4}
   10: {1,3}         48: {1,1,1,1,2}     87: {2,10}
   11: {5}           50: {1,3,3}         88: {1,1,1,5}
   12: {1,1,2}       54: {1,2,2,2}       95: {3,8}
   17: {7}           57: {2,8}           96: {1,1,1,1,1,2}
   18: {1,2,2}       59: {17}           100: {1,1,3,3}
   20: {1,1,3}       62: {1,11}         108: {1,1,2,2,2}
   21: {2,4}         63: {2,2,4}        109: {29}
   22: {1,5}         65: {3,6}          111: {2,12}
   24: {1,1,1,2}     67: {19}           114: {1,2,8}
   25: {3,3}         68: {1,1,7}        115: {3,9}
   27: {2,2,2}       69: {2,9}          117: {2,2,6}
   31: {11}          70: {1,3,4}        118: {1,17}
   34: {1,7}         72: {1,1,1,2,2}    119: {4,7}
   35: {3,4}         77: {4,5}          121: {5,5}
   36: {1,1,2,2}     78: {1,2,6}        124: {1,1,11}
   39: {2,6}         80: {1,1,1,1,3}    125: {3,3,3}

Crossrefs

These are numbers n such that A279952(n) = 1.
Prime-indexed primes are A006450, with products A076610.
The number of prime prime indices is A257994.
Numbers with at least one prime prime index are A331386.
The set S of numbers with exactly one prime index in S are A331785.
The set S of numbers with exactly one distinct prime index in S are A331913.
Numbers with at most one prime prime index are A331914.
Numbers with at most one distinct prime prime index are A331995.
Cf. A000040, A000720, A007097, A007821, A018252, A112798, A289509, A320628, A330944, A330945, A331784.

Programs

  • Mathematica
    Select[Range[100],Count[PrimePi/@First/@FactorInteger[#],_?PrimeQ]==1&]

A331995 Numbers with at most one distinct prime prime index.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 52, 53, 54, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 70, 71, 72, 73, 74, 76
Offset: 1

Views

Author

Gus Wiseman, Feb 08 2020

Keywords

Comments

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.

Examples

			The sequence of terms together with their prime indices begins:
   1: {}           22: {1,5}          44: {1,1,5}
   2: {1}          23: {9}            46: {1,9}
   3: {2}          24: {1,1,1,2}      47: {15}
   4: {1,1}        25: {3,3}          48: {1,1,1,1,2}
   5: {3}          26: {1,6}          49: {4,4}
   6: {1,2}        27: {2,2,2}        50: {1,3,3}
   7: {4}          28: {1,1,4}        52: {1,1,6}
   8: {1,1,1}      29: {10}           53: {16}
   9: {2,2}        31: {11}           54: {1,2,2,2}
  10: {1,3}        32: {1,1,1,1,1}    56: {1,1,1,4}
  11: {5}          34: {1,7}          57: {2,8}
  12: {1,1,2}      35: {3,4}          58: {1,10}
  13: {6}          36: {1,1,2,2}      59: {17}
  14: {1,4}        37: {12}           61: {18}
  16: {1,1,1,1}    38: {1,8}          62: {1,11}
  17: {7}          39: {2,6}          63: {2,2,4}
  18: {1,2,2}      40: {1,1,1,3}      64: {1,1,1,1,1,1}
  19: {8}          41: {13}           65: {3,6}
  20: {1,1,3}      42: {1,2,4}        67: {19}
  21: {2,4}        43: {14}           68: {1,1,7}

Crossrefs

These are numbers n such that A279952(n) <= 1.
Prime-indexed primes are A006450, with products A076610.
Numbers whose prime indices are not all prime are A330945.
Numbers with at least one prime prime index are A331386.
The set S of numbers with at most one prime index in S are A331784.
The set S of numbers with at most one distinct prime index in S are A331912.
Numbers with at most one prime prime index are A331914.
Numbers with exactly one prime prime index are A331915.
Numbers with exactly one distinct prime prime index are A331916.
Cf. A000040, A000720, A001221, A007097, A007821, A112798, A257994, A320628, A330944, A331785, A331912, A331913.

Programs

  • Mathematica
    Select[Range[100],Count[PrimePi/@First/@FactorInteger[#],_?PrimeQ]<=1&]
Showing 1-7 of 7 results.

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