A280996 -id:A280996 - cp's OEIS frontend

A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

2, 4, 5, 8, 10, 13, 16, 17, 20, 23, 25, 26, 31, 32, 34, 37, 40, 43, 46, 47, 50, 52, 61, 62, 64, 65, 67, 68, 73, 74, 79, 80, 85, 86, 89, 92, 94, 100, 103, 104, 107, 109, 113, 115, 122, 124, 125, 128, 130, 134, 136, 137, 146, 148, 149, 151, 155, 158, 160, 163

Offset: 1

Gus Wiseman, Aug 16 2018

nonn

A self-describing sequence.
The prime indices of m are the numbers k such that prime(k) divides m.
The sequence is monotonically increasing, since once a number is rejected it stays rejected. Sequence is closed under multiplication for a similar reason. - N. J. A. Sloane, Aug 26 2018

			After the initial term 2, the next term cannot be 3 because 3 has prime index 2, and 2 is already in the sequence. The next term could be 10, which has prime indices 1 and 3, but 4 (with prime index 1) is smaller. So a(2) = 4.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000002, A001462, A079000, A079254, A214577, A276625, A277098, A280996, A303431.

For first differences see A317963, for primes see A317964.

A:= NULL:
P:= {}:
for n  from 2 to 1000 do
  pn:= numtheory:-factorset(n);
  if pn intersect P = {} then
    A:= A, n;
    P:= P union {ithprime(n)};
  fi
od:
A; # Robert Israel, Aug 26 2018

gaQ[n_]:=Or[n==0,And@@Cases[FactorInteger[n],{p_,k_}:>!gaQ[PrimePi[p]]]];
Select[Range[100],gaQ]

Added "infinite" to definition. - N. J. A. Sloane, Sep 28 2019

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

A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

Original entry on oeis.org

1, 4, 8, 16, 32, 49, 64, 128, 256, 343, 361, 512, 1024, 2048, 2401, 2809, 4096, 6859, 8192, 16384, 16807, 17161, 32768, 51529, 65536, 96721, 117649, 130321, 131072, 148877, 262144, 516961, 524288, 823543, 1048576, 2097152, 2248091, 2476099, 2621161, 4194304

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

Lone-child-avoiding means there are no unary branchings.
In an achiral rooted tree, the branches of any given vertex are all equal.
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 one and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.

			The sequence of all lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
      1: o
      4: (oo)
      8: (ooo)
     16: (oooo)
     32: (ooooo)
     49: ((oo)(oo))
     64: (oooooo)
    128: (ooooooo)
    256: (oooooooo)
    343: ((oo)(oo)(oo))
    361: ((ooo)(ooo))
    512: (ooooooooo)
   1024: (oooooooooo)
   2048: (ooooooooooo)
   2401: ((oo)(oo)(oo)(oo))
   2809: ((oooo)(oooo))
   4096: (oooooooooooo)
   6859: ((ooo)(ooo)(ooo))
   8192: (ooooooooooooo)
  16384: (oooooooooooooo)
  16807: ((oo)(oo)(oo)(oo)(oo))
  17161: ((ooooo)(ooooo))
  32768: (ooooooooooooooo)
  51529: (((oo)(oo))((oo)(oo)))
  65536: (oooooooooooooooo)
  96721: ((oooooo)(oooooo))

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The first 30 lone-child-avoiding achiral rooted trees.

A subset of A025475 (nonprime prime powers).
The enumeration of these trees by vertices is A167865.
Not requiring lone-child-avoidance gives A214577.
The semi-achiral version is A320269.
The semi-lone-child-avoiding version is A331992.
Achiral rooted trees are counted by A003238.
MG-numbers of planted achiral rooted trees are A280996.
MG-numbers of lone-child-avoiding rooted trees are A291636.
Cf. A001678, A007097, A061775, A196050, A276625, A291441, A320230, A320268, A331912, A331936, A331963, A331965, A331966, A331991.

msQ[n_]:=n==1||!PrimeQ[n]&&PrimePowerQ[n]&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Intersection of A214577 (achiral) and A291636 (lone-child-avoiding).

A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.

Original entry on oeis.org

1, 2, 4, 8, 9, 16, 27, 32, 49, 64, 81, 128, 243, 256, 343, 361, 512, 529, 729, 1024, 2048, 2187, 2401, 2809, 4096, 6561, 6859, 8192, 10609, 12167, 16384, 16807, 17161, 19683, 32768, 51529, 59049, 65536, 96721, 117649, 130321, 131072, 148877, 175561, 177147

Offset: 1

Gus Wiseman, Feb 06 2020

nonn

A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless that child is an endpoint/leaf.
In an achiral rooted tree, the branches of any given vertex are all equal.
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 one, two, and all numbers of the form prime(j)^k where k > 1 and j is already in the sequence.

			The sequence of all semi-lone-child-avoiding achiral rooted trees together with their Matula-Goebel numbers begins:
     1: o
     2: (o)
     4: (oo)
     8: (ooo)
     9: ((o)(o))
    16: (oooo)
    27: ((o)(o)(o))
    32: (ooooo)
    49: ((oo)(oo))
    64: (oooooo)
    81: ((o)(o)(o)(o))
   128: (ooooooo)
   243: ((o)(o)(o)(o)(o))
   256: (oooooooo)
   343: ((oo)(oo)(oo))
   361: ((ooo)(ooo))
   512: (ooooooooo)
   529: (((o)(o))((o)(o)))
   729: ((o)(o)(o)(o)(o)(o))
  1024: (oooooooooo)

Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Gus Wiseman, The first 36 semi-lone-child-avoiding achiral rooted trees.

Except for two, a subset of A025475 (nonprime prime powers).
Not requiring achirality gives A331935.
The semi-achiral version is A331936.
The fully-chiral version is A331963.
The semi-chiral version is A331994.
The non-semi version is counted by A331967.
The enumeration of these trees by vertices is A331991.
Achiral rooted trees are counted by A003238.
MG-numbers of achiral rooted trees are A214577.
Cf. A001678, A007097, A050381, A061775, A167865, A196050, A276625, A280996, A291441, A291636, A320230, A320269, A331912, A331933, A331965.

msQ[n_]:=n<=2||!PrimeQ[n]&&Length[FactorInteger[n]]<=1&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[10000],msQ]

Intersection of A214577 (achiral) and A331935 (semi-lone-child-avoiding).

A322385 2 and prime numbers whose prime index is a product of at least two not necessarily distinct prime numbers already in the sequence.

Original entry on oeis.org

2, 7, 19, 43, 53, 107, 131, 163, 227, 263, 311, 383, 443, 521, 577, 613, 719, 751, 881, 1021, 1193, 1301, 1307, 1423, 1619, 1667, 1699, 1993, 2003, 2161, 2309, 2311, 2437, 2539, 2693, 2939, 2969, 3167, 3209, 3671, 3767, 3779, 3833, 4423, 4481, 4597, 4871, 5147

Offset: 1

Gus Wiseman, Dec 05 2018

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.

			We have 1993 = prime(301) = prime(7 * 43). Since 7 and 43 already belong to the sequence, so does 1993.

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000002, A001462, A007097, A079000, A079254, A277098, A280996, A291636, A304360, A320628, A322386.

ppQ[n_]:=And[PrimeQ[n],!PrimeQ[PrimePi[n]],And@@ppQ/@First/@If[n==2,{},FactorInteger[PrimePi[n]]]];
Select[Range[1000],ppQ]

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A304360 Lexicographically earliest infinite sequence of numbers m > 1 with the property that none of the prime indices of m are in the sequence.

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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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A331967 Matula-Goebel numbers of lone-child-avoiding achiral rooted trees.

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A331992 Matula-Goebel numbers of semi-lone-child-avoiding achiral rooted trees.

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A322385 2 and prime numbers whose prime index is a product of at least two not necessarily distinct prime numbers already in the sequence.

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