A331916 -id:A331916 - cp's OEIS frontend

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

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

Gus Wiseman, Feb 08 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.

			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}

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.

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

A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 152, 172, 196, 212, 214, 224, 256, 262, 304, 326, 343, 344, 361, 392, 424, 428, 448, 454, 512, 524, 526, 608, 622, 652, 686, 688, 722, 766, 784, 848, 856, 886, 896, 908, 1024, 1042, 1048, 1052

Offset: 1

Gus Wiseman, Oct 08 2018

nonn

First differs from A331871 in lacking 1589.
Lone-child-avoiding means there are no unary branchings.
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.

			The sequence of rooted trees together with their Matula-Goebel numbers begins:
    1: o
    4: (oo)
    8: (ooo)
   14: (o(oo))
   16: (oooo)
   28: (oo(oo))
   32: (ooooo)
   38: (o(ooo))
   49: ((oo)(oo))
   56: (ooo(oo))
   64: (oooooo)
   76: (oo(ooo))
   86: (o(o(oo)))
   98: (o(oo)(oo))
  106: (o(oooo))
  112: (oooo(oo))
  128: (ooooooo)
  152: (ooo(ooo))
  172: (oo(o(oo)))
  196: (oo(oo)(oo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.

Cf. A002541, A070776, A167865, A214577, A317710, A320222, A320226.
The same-tree version is A291441.
Not requiring lone-child-avoidance gives A320230.
The enumeration of these trees by vertices is A320268.
The semi-lone-child-avoiding version is A331936.
If the non-leaf branches are all different instead of equal we get A331965.
The fully-achiral case is A331967.
Achiral rooted trees are counted by A003238.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A061775, A196050, A276625, A280996, A306202, A331912, A331916, A331966.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]
hmakQ[n_]:=And[!PrimeQ[n],SameQ@@DeleteCases[primeMS[n],1],And@@hmakQ/@primeMS[n]];Select[Range[1000],hmakQ[#]&]

Updated with corrected terminology by Gus Wiseman, Feb 06 2020

A331785 Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.

Original entry on oeis.org

1, 2, 3, 5, 11, 14, 21, 26, 31, 34, 35, 38, 39, 43, 46, 51, 57, 58, 65, 69, 73, 74, 77, 82, 85, 87, 94, 95, 98, 101, 106, 111, 115, 118, 122, 123, 127, 134, 139, 141, 142, 143, 145, 147, 149, 158, 159, 163, 166, 167, 177, 178, 182, 183, 185, 187, 191, 194, 199

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.

			The sequence of terms together with their prime indices begins:
    1: {}         73: {21}       142: {1,20}     205: {3,13}
    2: {1}        74: {1,12}     143: {5,6}      206: {1,27}
    3: {2}        77: {4,5}      145: {3,10}     209: {5,8}
    5: {3}        82: {1,13}     147: {2,4,4}    213: {2,20}
   11: {5}        85: {3,7}      149: {35}       214: {1,28}
   14: {1,4}      87: {2,10}     158: {1,22}     217: {4,11}
   21: {2,4}      94: {1,15}     159: {2,16}     218: {1,29}
   26: {1,6}      95: {3,8}      163: {38}       226: {1,30}
   31: {11}       98: {1,4,4}    166: {1,23}     233: {51}
   34: {1,7}     101: {26}       167: {39}       235: {3,15}
   35: {3,4}     106: {1,16}     177: {2,17}     237: {2,22}
   38: {1,8}     111: {2,12}     178: {1,24}     238: {1,4,7}
   39: {2,6}     115: {3,9}      182: {1,4,6}    245: {3,4,4}
   43: {14}      118: {1,17}     183: {2,18}     249: {2,23}
   46: {1,9}     122: {1,18}     185: {3,12}     253: {5,9}
   51: {2,7}     123: {2,13}     187: {5,7}      262: {1,32}
   57: {2,8}     127: {31}       191: {43}       265: {3,16}
   58: {1,10}    134: {1,19}     194: {1,25}     266: {1,4,8}
   65: {3,6}     139: {34}       199: {46}       267: {2,24}
   69: {2,9}     141: {2,15}     201: {2,19}     269: {57}
For example, the prime indices of 77 are {4,5}, of which only 5 is in the sequence, so 77 is in the sequence.

Closed under A000040.
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 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, A331915, A331916.

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

A331914 Numbers with at most one prime prime index, counted with multiplicity.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 13, 14, 16, 17, 19, 20, 21, 22, 23, 24, 26, 28, 29, 31, 32, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 46, 47, 48, 49, 52, 53, 56, 57, 58, 59, 61, 62, 64, 65, 67, 68, 69, 70, 71, 73, 74, 76, 77, 78, 79, 80, 82, 83, 84, 86, 87

Offset: 1

Gus Wiseman, Feb 08 2020

nonn

First differs from A324935 in having 39.

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: {}           24: {1,1,1,2}      52: {1,1,6}
   2: {1}          26: {1,6}          53: {16}
   3: {2}          28: {1,1,4}        56: {1,1,1,4}
   4: {1,1}        29: {10}           57: {2,8}
   5: {3}          31: {11}           58: {1,10}
   6: {1,2}        32: {1,1,1,1,1}    59: {17}
   7: {4}          34: {1,7}          61: {18}
   8: {1,1,1}      35: {3,4}          62: {1,11}
  10: {1,3}        37: {12}           64: {1,1,1,1,1,1}
  11: {5}          38: {1,8}          65: {3,6}
  12: {1,1,2}      39: {2,6}          67: {19}
  13: {6}          40: {1,1,1,3}      68: {1,1,7}
  14: {1,4}        41: {13}           69: {2,9}
  16: {1,1,1,1}    42: {1,2,4}        70: {1,3,4}
  17: {7}          43: {14}           71: {20}
  19: {8}          44: {1,1,5}        73: {21}
  20: {1,1,3}      46: {1,9}          74: {1,12}
  21: {2,4}        47: {15}           76: {1,1,8}
  22: {1,5}        48: {1,1,1,1,2}    77: {4,5}
  23: {9}          49: {4,4}          78: {1,2,6}

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 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 exactly one prime prime index are A331915.
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, A018252, A112798, A320628, A330945, A331785.

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

A331913 Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 16, 17, 19, 23, 25, 26, 27, 31, 32, 39, 49, 52, 53, 58, 59, 64, 65, 67, 74, 81, 82, 83, 86, 87, 91, 94, 97, 101, 103, 104, 111, 116, 117, 121, 122, 123, 125, 127, 128, 129, 131, 141, 142, 143, 145, 146, 148, 158, 164, 167, 172, 178

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.

			The sequence of terms together with their prime indices begins:
    1: {}              52: {1,1,6}          116: {1,1,10}
    2: {1}             53: {16}             117: {2,2,6}
    3: {2}             58: {1,10}           121: {5,5}
    4: {1,1}           59: {17}             122: {1,18}
    5: {3}             64: {1,1,1,1,1,1}    123: {2,13}
    7: {4}             65: {3,6}            125: {3,3,3}
    8: {1,1,1}         67: {19}             127: {31}
    9: {2,2}           74: {1,12}           128: {1,1,1,1,1,1,1}
   11: {5}             81: {2,2,2,2}        129: {2,14}
   16: {1,1,1,1}       82: {1,13}           131: {32}
   17: {7}             83: {23}             141: {2,15}
   19: {8}             86: {1,14}           142: {1,20}
   23: {9}             87: {2,10}           143: {5,6}
   25: {3,3}           91: {4,6}            145: {3,10}
   26: {1,6}           94: {1,15}           146: {1,21}
   27: {2,2,2}         97: {25}             148: {1,1,12}
   31: {11}           101: {26}             158: {1,22}
   32: {1,1,1,1,1}    103: {27}             164: {1,1,13}
   39: {2,6}          104: {1,1,1,6}        167: {39}
   49: {4,4}          111: {2,12}           172: {1,1,14}

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 at most one distinct prime index in S are A331912.
Cf. A000002, A000720, A001222, A001462, A324696, A331683, A331873, A331915, A331916.

aQ[n_]:=n==1||Length[Select[PrimePi/@First/@FactorInteger[n],aQ]]==1;
Select[Range[200],aQ]

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

Gus Wiseman, Feb 08 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.

			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}

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.

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

cp's OEIS Frontend

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

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A320269 Matula-Goebel numbers of lone-child-avoiding rooted trees in which the non-leaf branches directly under any given node are all equal (semi-achirality).

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A331785 Lexicographically earliest sequence containing 1 and all positive integers with exactly one prime index already in the sequence, counting multiplicity.

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A331914 Numbers with at most one prime prime index, counted with multiplicity.

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A331913 Lexicographically earliest sequence containing 1 and all positive integers that have exactly one distinct prime index already in the sequence.

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A331995 Numbers with at most one distinct prime prime index.

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