A276625 -id:A276625 - 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

A290822 Transitive numbers: Matula-Goebel numbers of transitive rooted trees.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 14, 16, 18, 24, 28, 30, 32, 36, 38, 42, 48, 54, 56, 60, 64, 72, 76, 78, 84, 90, 96, 98, 106, 108, 112, 114, 120, 126, 128, 138, 144, 150, 152, 156, 162, 168, 180, 192, 196, 210, 212, 216, 222, 224, 228, 234, 238, 240, 252, 256, 262, 266, 270

Offset: 1

Gus Wiseman, Oct 19 2017

nonn

A number x is transitive if whenever prime(y) divides x and prime(z) divides y, we have prime(z) divides x.

			The sequence of transitive rooted trees begins:
1  o
2  (o)
4  (oo)
6  (o(o))
8  (ooo)
12 (oo(o))
14 (o(oo))
16 (oooo)
18 (o(o)(o))
24 (ooo(o))
28 (oo(oo))
30 (o(o)((o)))
32 (ooooo)
36 (oo(o)(o))
38 (o(ooo))
42 (o(o)(oo))
48 (oooo(o))
54 (o(o)(o)(o))
56 (ooo(oo))
60 (oo(o)((o)))
64 (oooooo)
72 (ooo(o)(o))
76 (oo(ooo))
78 (o(o)(o(o)))
84 (oo(o)(oo))
90 (o(o)(o)((o)))
96 (ooooo(o))
98 (o(oo)(oo))

Robert P. P. McKone, Table of n, a(n) for n = 1..9999

Cf. A007097, A276625, A279861, A290689, A290760.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
Select[Range[270],Divisible[#,Times@@subprimes[#]]&]

A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

Original entry on oeis.org

1, 3, 7, 9, 11, 13, 19, 21, 27, 29, 33, 37, 39, 43, 47, 49, 53, 57, 59, 61, 63, 71, 77, 79, 81, 83, 87, 89, 91, 97, 99, 101, 107, 111, 113, 117, 121, 127, 129, 131, 133, 139, 141, 143, 147, 149, 151, 159, 163, 169, 171, 173, 177, 179, 181, 183, 189, 193, 197

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}
  11: {5}
  13: {6}
  19: {8}
  21: {2,4}
  27: {2,2,2}
  29: {10}
  33: {2,5}
  37: {12}
  39: {2,6}
  43: {14}
  47: {15}
  49: {4,4}
  53: {16}
  57: {2,8}
  59: {17}
  61: {18}
  63: {2,2,4}

Complement of A324694. Prime indices are A304360.
Cf. A000002, A000720, A001222, A001462, A007097, A055396, A061395, A079000, A079254, A109298, A112798, A276625, A277098.
Cf. A324696, A324697, A324698, A324699, A324700, A324701, A324702, A324703, A324704, A324705.

aQ[n_]:=And@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>!aQ[PrimePi[p]]];
Select[Range[100],aQ]

A306844 Number of anti-transitive rooted trees with n nodes.

Original entry on oeis.org

1, 1, 2, 3, 7, 14, 36, 83, 212, 532, 1379, 3577, 9444, 25019, 66943, 179994, 487031, 1323706, 3614622, 9907911

Offset: 1

Gus Wiseman, Mar 13 2019

A rooted tree is anti-transitive if the subbranches are disjoint from the branches, i.e., no branch of a branch is a branch.

			The a(1) = 1 through a(6) = 14 anti-transitive rooted trees:
  o  (o)  (oo)   (ooo)    (oooo)     (ooooo)
          ((o))  ((oo))   ((ooo))    ((oooo))
                 (((o)))  (((oo)))   (((ooo)))
                          ((o)(o))   ((o)(oo))
                          ((o(o)))   ((o(oo)))
                          (o((o)))   ((oo(o)))
                          ((((o))))  (o((oo)))
                                     (oo((o)))
                                     ((((oo))))
                                     (((o)(o)))
                                     (((o(o))))
                                     ((o((o))))
                                     (o(((o))))
                                     (((((o)))))

Gus Wiseman, The a(7) = 36 anti-transitive rooted trees.
Gus Wiseman, The a(10) = 532 anti-transitive rooted trees.

Cf. A276625, A279861, A279861, A290689, A290760, A304360.

Cf. A324694, A324695, A324738, A324741, A324743, A324751, A324754, A324756, A324758, A324759, A324764.

rtall[n_]:=Union[Sort/@Join@@(Tuples[rtall/@#]&/@IntegerPartitions[n-1])];
Table[Length[Select[rtall[n],Intersection[Union@@#,#]=={}&]],{n,10}]

a(16)-a(20) from Jinyuan Wang, Jun 20 2020

A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.

Original entry on oeis.org

1, 4, 8, 14, 16, 28, 32, 38, 49, 56, 64, 76, 86, 98, 106, 112, 128, 133, 152, 172, 196, 212, 214, 224, 256, 262, 266, 301, 304, 326, 343, 344, 361, 371, 392, 424, 428, 448, 454, 512, 524, 526, 532, 602, 608, 622, 652, 686, 688, 722, 742, 749, 766, 784, 817

Offset: 1

Gus Wiseman, Aug 28 2017

nonn

We say that a rooted tree is lone-child-avoiding if no vertex has exactly one child.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches. This gives a bijective correspondence between positive integers and unlabeled rooted trees.
An alternative definition: n is in the sequence iff n is 1 or the product of two or more not necessarily distinct prime numbers whose prime indices already belong to the sequence. For example, 14 is in the sequence because 14 = prime(1) * prime(4) and 1 and 4 both already belong to the sequence.

			The sequence of all lone-child-avoiding 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)
  133: ((oo)(ooo))
  152: (ooo(ooo))
  172: (oo(o(oo)))

David Callan, A sign-reversing involution to count labeled lone-child-avoiding trees, arXiv:1406.7784 [math.CO], (30-June-2014).
Gus Wiseman, Sequences counting series-reduced and lone-child-avoiding trees by number of vertices.
Index entries for sequences related to Matula-Goebel numbers

These trees are counted by A001678.
The case with more than two branches is A331490.
Unlabeled rooted trees are counted by A000081.
Topologically series-reduced rooted trees are counted by A001679.
Labeled lone-child-avoiding rooted trees are counted by A060356.
Labeled lone-child-avoiding unrooted trees are counted by A108919.
MG numbers of singleton-reduced rooted trees are A330943.
MG numbers of topologically series-reduced rooted trees are A331489.
Cf. A007097, A061775, A109082, A109129, A111299, A196050, A198518, A276625, A291441, A291442, A331488.

nn=2000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
srQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]>1,And@@srQ/@m]]];
Select[Range[nn],srQ]

Updated with corrected terminology by Gus Wiseman, Jan 20 2020

A109298 Primal codes of finite idempotent functions on positive integers.

Original entry on oeis.org

1, 2, 9, 18, 125, 250, 1125, 2250, 2401, 4802, 21609, 43218, 161051, 300125, 322102, 600250, 1449459, 2701125, 2898918, 4826809, 5402250, 9653618, 20131375, 40262750, 43441281, 86882562, 181182375, 362364750, 386683451, 410338673, 603351125, 773366902, 820677346

Offset: 1

Jon Awbrey, Jul 06 2005

nonn

Finite idempotent functions are identity maps on finite subsets, counting the empty function as the idempotent on the empty set.
From Gus Wiseman, Mar 09 2019: (Start)
Also numbers whose ordered prime signature is equal to the distinct prime indices in increasing order. A prime index of n is a number m such that prime(m) divides n. The ordered prime signature (A124010) is the sequence of multiplicities (or exponents) in a number's prime factorization, taken in order of the prime base. The case where the prime indices are taken in decreasing order is A324571.
Also numbers divisible by prime(k) exactly k times for each prime index k. These are a kind of self-describing numbers (cf. A001462, A304679).
Also Heinz numbers of integer partitions where the multiplicity of m is m for all m in the support (counted by A033461). The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
Also products of distinct elements of A062457. For example, 43218 = prime(1)^1 * prime(2)^2 * prime(4)^4.
(End)

			Writing (prime(i))^j as i:j, we have the following table of examples:
Primal Codes of Finite Idempotent Functions on Positive Integers
` ` ` 1 = { }
` ` ` 2 = 1:1
` ` ` 9 = ` ` 2:2
` ` `18 = 1:1 2:2
` ` 125 = ` ` ` ` 3:3
` ` 250 = 1:1 ` ` 3:3
` `1125 = ` ` 2:2 3:3
` `2250 = 1:1 2:2 3:3
` `2401 = ` ` ` ` ` ` 4:4
` `4802 = 1:1 ` ` ` ` 4:4
` 21609 = ` ` 2:2 ` ` 4:4
` 43218 = 1:1 2:2 ` ` 4:4
`161051 = ` ` ` ` ` ` ` ` 5:5
`300125 = ` ` ` ` 3:3 4:4
`322102 = 1:1 ` ` ` ` ` ` 5:5
`600250 = 1:1 ` ` 3:3 4:4
From _Gus Wiseman_, Mar 09 2019: (Start)
The sequence of terms together with their prime indices begins as follows. For example, we have 18: {1,2,2} because 18 = prime(1) * prime(2) * prime(2) has prime signature {1,2} and the distinct prime indices are also {1,2}.
       1: {}
       2: {1}
       9: {2,2}
      18: {1,2,2}
     125: {3,3,3}
     250: {1,3,3,3}
    1125: {2,2,3,3,3}
    2250: {1,2,2,3,3,3}
    2401: {4,4,4,4}
    4802: {1,4,4,4,4}
   21609: {2,2,4,4,4,4}
   43218: {1,2,2,4,4,4,4}
  161051: {5,5,5,5,5}
  300125: {3,3,3,4,4,4,4}
  322102: {1,5,5,5,5,5}
  600250: {1,3,3,3,4,4,4,4}
(End)

David A. Corneth, Table of n, a(n) for n = 1..10000
J. Awbrey, Riffs and Rotes

Cf. A076954, A106177, A108352, A108371, A109297, A109301.
Cf. A001156, A033461, A056239, A062457, A112798, A118914, A124010 (ordered prime signature), A181819, A276078, A304679.
Cf. A324524, A324525, A324570, A324571, A324572, A324587, A324588.
Sequences related to self-description: A000002, A001462, A079000, A079254, A276625, A304360.

Select[Range[10000],And@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]==k]&]

is(n) = my(f = factor(n)); for(i = 1, #f~, if(prime(f[i, 2]) != f[i, 1], return(0))); 1 \\ David A. Corneth, Mar 09 2019

Sum_{n>=1} 1/a(n) = Product_{n>=1} (1 + 1/prime(n)^n) = 1.6807104966... - Amiram Eldar, Jan 03 2021

Offset set to 1, missing terms inserted and more terms added by Alois P. Heinz, Mar 08 2019

A301700 Number of aperiodic rooted trees with n nodes.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 21, 52, 120, 290, 697, 1713, 4200, 10446, 26053, 65473, 165257, 419357, 1068239, 2732509, 7013242, 18059960, 46641983, 120790324, 313593621, 816046050, 2128101601, 5560829666, 14557746453, 38177226541, 100281484375, 263815322761, 695027102020

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

An unlabeled rooted tree is aperiodic if the multiset of branches of the root is an aperiodic multiset, meaning it has relatively prime multiplicities, and each branch is also aperiodic.

			The a(6) = 10 aperiodic trees are (((((o))))), (((o(o)))), ((o((o)))), ((oo(o))), (o(((o)))), (o(o(o))), ((o)((o))), (oo((o))), (o(o)(o)), (ooo(o)).

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A000081, A000740, A000837, A001678, A003238, A004111, A007716, A007916, A100953, A276625, A284639, A290689, A298422, A303386, A303431.

arut[n_]:=arut[n]=If[n===1,{{}},Join@@Function[c,Select[Union[Sort/@Tuples[arut/@c]],GCD@@Length/@Split[#]===1&]]/@IntegerPartitions[n-1]];
Table[Length[arut[n]],{n,20}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v,vector(#v,n,1/n))))-1, -#v)}
MoebiusT(v)={vector(#v, n, sumdiv(n,d,moebius(n/d)*v[d]))}
seq(n)={my(v=[1]); for(n=2, n, v=concat([1], MoebiusT(EulerT(v)))); v} \\ Andrew Howroyd, Sep 01 2018

Terms a(21) and beyond from Andrew Howroyd, Sep 01 2018

A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3).

Original entry on oeis.org

4, 14, 49, 86, 301, 454, 886, 1589, 1849, 3101, 3986, 6418, 9761, 13766, 13951, 19049, 22463, 26798, 31754, 48181, 51529, 57026, 75266, 85699, 93793, 100561, 111139, 128074, 137987, 196249, 199591, 203878, 263431, 295969, 298154, 302426, 426058, 448259, 452411

Offset: 1

Keith Briggs, Nov 02 2005

nonn

This sequence should probably start with 1. Then a number k is in the sequence iff k = 1 or k = prime(x) * prime(y) with x and y already in the sequence. - Gus Wiseman, May 04 2021

			From _Gus Wiseman_, May 04 2021: (Start)
The sequence of trees (starting with 1) begins:
     1: o
     4: (oo)
    14: (o(oo))
    49: ((oo)(oo))
    86: (o(o(oo)))
   301: ((oo)(o(oo)))
   454: (o((oo)(oo)))
   886: (o(o(o(oo))))
  1589: ((oo)((oo)(oo)))
  1849: ((o(oo))(o(oo)))
  3101: ((oo)(o(o(oo))))
  3986: (o((oo)(o(oo))))
  6418: (o(o((oo)(oo))))
  9761: ((o(oo))((oo)(oo)))
(End)

Kevin Ryde, Table of n, a(n) for n = 1..5000 (terms 1..186 from Charles R Greathouse IV)
Keith Briggs, Matula numbers and rooted trees
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Kevin Ryde, PARI/GP Code
Index entries for sequences related to Matula-Goebel numbers

Cf. A245824 (by number of leaves).
Cf. A005517, A005518, A061773, A245824.
These trees are counted by 2*A001190 - 1.
The semi-binary version is A292050 (counted by A001190).
The semi-identity case is A339193 (counted by A063895).
A000081 counts unlabeled rooted trees with n nodes.
A007097 ranks rooted chains.
A276625 ranks identity trees, counted by A004111.
A306202 ranks semi-identity trees, counted by A306200.
A306203 ranks balanced semi-identity trees, counted by A306201.
A331965 ranks lone-child avoiding semi-identity trees, counted by A331966.
Cf. A036656, A061775, A196050, A291636, A320230, A331935, A331965.

nn=20000;
primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
binQ[n_]:=Or[n===1,With[{m=primeMS[n]},And[Length[m]===2,And@@binQ/@m]]];
Select[Range[2,nn],binQ] (* Gus Wiseman, Aug 28 2017 *)

i(n)=n==2 || is(primepi(n))
is(n)=if(n<14,return(n==4)); my(f=factor(n),t=#f[,1]); if(t>1, t==2 && f[1,2]==1 && f[2,2]==1 && i(f[1,1]) && i(f[2,1]), f[1,2]==2 && i(f[1,1])) \\ Charles R Greathouse IV, Mar 29 2013

list(lim)=my(v=List(), t); forprime(p=2, sqrt(lim), t=p; forprime(q=p, lim\t, if(i(p)&&i(q), listput(v, t*q)))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Mar 29 2013

PARI
```
\\ Also see links.
```

The Matula tree of k is defined as follows:
matula(k):
    create a node labeled k
    for each prime factor m of k:
       add the subtree matula(prime(m)), by an edge labeled m
    return the node

Definition corrected by Charles R Greathouse IV, Mar 29 2013

a(27)-a(39) from Charles R Greathouse IV, Mar 29 2013

A277098 Finitary primes. Primes of finitary index.

Original entry on oeis.org

2, 3, 5, 11, 13, 29, 31, 41, 47, 79, 101, 109, 113, 127, 137, 167, 179, 211, 257, 271, 293, 313, 317, 397, 401, 421, 449, 487, 491, 547, 599, 601, 617, 677, 709, 733, 773, 811, 823, 829, 907, 929, 977, 991, 1033, 1063, 1109, 1187, 1231, 1259, 1297, 1361, 1429, 1483, 1489, 1559, 1609, 1621, 1741, 1759, 1831, 1871, 1889

Offset: 1

Gus Wiseman, Sep 29 2016

nonn

Definition: prime(n) is a finitary prime iff n is a product of distinct finitary primes, where prime = A000040. This sequence could be described as a "multiplicative Aronson transform" of A005117. (Aronson transforms such as A079000 satisfy "n is in a if and only if a(n) is in b".

The composite bijection (finitary primes -> finitary numbers -> finite sets of finitary primes) can be used to construct a natural linear extension SET : N -> F where F is the partially ordered inverse limit of all finite Boolean algebras of finite sets of positive integers. Then a(n) = prime(Prod_p a(p)) where the product is over SET(n).

			The sequence of all nonempty finite sets of positive integers (a=1 b=2.. *=27) begins:
0,a,b,c,ab,ac,d,e,bc,ad,ae,f,abc,
g,bd,be,h,i,cd,af,ag,ce,abd,abe,
j,ah,bf,bg,ai,k,l,acd,m,bh,n,ace,
o,bi,de,cf,cg,aj,bcd,p,abf,q,abg,
bce,ak,ch,r,al,am,ci,bj,abh,an,s,
t,ao,abi,ade,acf,u,bk,acg,v,w,df,
bl,abcd,ap,bm,dg,aq,ef,bn,abce,
cj,x,y,eg,ach,bo,z,ar,bde,bcf,*

Gus Wiseman, Table of n, a(n) for n = 1..10000

Subsequence of A302491.

Cf. A000040, A005117, A079000, A079254, A276625.

has(p)=if(p<7, 1, my(f=factor(primepi(p))); if(vecmax(f[,2])>1, return(0)); for(i=1,#f~, if(!has(f[i,1]), return(0))); 1)
is(n)=isprime(n) && has(n) \\ Charles R Greathouse IV, Aug 03 2023

a(n) = A000040(A276625(n)).

A303431 Aperiodic tree numbers. Matula-Goebel numbers of aperiodic rooted trees.

Original entry on oeis.org

1, 2, 3, 5, 6, 10, 11, 12, 13, 15, 18, 20, 22, 24, 26, 29, 30, 31, 33, 37, 39, 40, 41, 44, 45, 47, 48, 50, 52, 54, 55, 58, 60, 61, 62, 65, 66, 71, 72, 74, 75, 78, 79, 80, 82, 87, 88, 89, 90, 93, 94, 96, 99, 101, 104, 108, 109, 110, 111, 113, 116, 117, 120, 122

Offset: 1

Gus Wiseman, Apr 23 2018

nonn

A positive integer is an aperiodic tree number iff either it is equal to 1 or it belongs to A007916 (numbers that are not perfect powers, or numbers whose prime multiplicities are relatively prime) and all of its prime indices are also aperiodic tree numbers, where a prime index of n is a number m such that prime(m) divides n.

			Sequence of aperiodic rooted trees begins:
01 o
02 (o)
03 ((o))
05 (((o)))
06 (o(o))
10 (o((o)))
11 ((((o))))
12 (oo(o))
13 ((o(o)))
15 ((o)((o)))
18 (o(o)(o))
20 (oo((o)))
22 (o(((o))))
24 (ooo(o))
26 (o(o(o)))
29 ((o((o))))
30 (o(o)((o)))
31 (((((o)))))
33 ((o)(((o))))

Cf. A000081, A000740, A000837, A007097, A007916, A052409, A052410, A061775, A111299, A214577, A275024, A276625, A290760, A290822, A291442, A298533, A298536, A301700, A302242, A303386.

zapQ[1]:=True;zapQ[n_]:=And[GCD@@FactorInteger[n][[All,2]]===1,And@@zapQ/@PrimePi/@FactorInteger[n][[All,1]]];
Select[Range[100],zapQ]

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.

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A290822 Transitive numbers: Matula-Goebel numbers of transitive rooted trees.

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A324695 Lexicographically earliest sequence of positive integers whose prime indices are not already in the sequence.

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Mathematica

A306844 Number of anti-transitive rooted trees with n nodes.

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A291636 Matula-Goebel numbers of lone-child-avoiding rooted trees.

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A109298 Primal codes of finite idempotent functions on positive integers.

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A301700 Number of aperiodic rooted trees with n nodes.

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A111299 Numbers whose Matula tree is a binary tree (i.e., root has degree 2 and all nodes except root and leaves have degree 3).