A007097 -id:A007097 - cp's OEIS frontend

A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

0, 1, 2, 0, 3, 0, 1, 0, 0, 0, 4, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 5, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 1, 0, 0, 0, 0, 0, 2, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0

Offset: 1

Henry Bottomley, Dec 31 2002

nonn

Fernandez calls this the order of primeness of n.
a(A007097(n))=n, for any n >= 0. - Paul Tek, Nov 12 2013
When a nonoriented rooted tree is encoded as a Matula-Goebel number n, a(n) tells how many edges needs to be climbed up from the root of the tree until the first branching vertex (or the top of the tree, if n is one of the terms of A007097) is encountered. Please see illustrations at A061773. - Antti Karttunen, Jan 27 2014
Zero-based column index of n in the Kimberling-style dispersion table of the primes (see A114537). - Allan C. Wechsler, Jan 09 2024

			a(1) = 0 since 1 is not prime;
a(2) = a(prime(1)) = a(1) + 1 = 1 + 0 = 1;
a(3) = a(prime(2)) = a(2) + 1 = 1 + 1 = 2;
a(4) = 0 since 4 is not prime;
a(5) = a(prime(3)) = a(3) + 1 = 2 + 1 = 3;
a(6) = 0 since 6 is not prime;
a(7) = a(prime(4)) = a(4) + 1 = 0 + 1 = 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
N. Fernandez, An order of primeness, F(p)
N. Fernandez, An order of primeness [cached copy, included with permission of the author]
Index entries for sequences related to Matula-Goebel numbers

A left inverse of A007097.
One less than A049076.
a(A000040(n)) = A049076(n).
Cf. A373338 (mod 2), A018252 (positions of zeros).
Cf. A000720, A049084, A061773.
Cf. permutations A235489, A250247/A250248, A250249/A250250, A245821/A245822 that all preserve a(n).
Cf. also array A114537 (A138947) and permutations A135141/A227413, A246681.

a078442 n = fst $ until ((== 0) . snd)
                        (\(i, p) -> (i + 1, a049084 p)) (-2, a000040 n)
-- Reinhard Zumkeller, Jul 14 2013

A078442 := proc(n)
    if not isprime(n) then
        0 ;
    else
        1+procname(numtheory[pi](n)) ;
    end if;
end proc: # R. J. Mathar, Jul 07 2012

a[n_] := a[n] = If[!PrimeQ[n], 0, 1+a[PrimePi[n]]]; Array[a, 105] (* Jean-François Alcover, Jan 26 2018 *)

A078442(n)=for(i=0,n, isprime(n) || return(i); n=primepi(n)) \\ M. F. Hasler, Mar 09 2010

a(n) = A049076(n)-1.
a(n) = if A049084(n) = 0 then 0 else a(A049084(n)) + 1. - Reinhard Zumkeller, Jul 14 2013
For all n, a(n) = A007814(A135141(n)) and a(A227413(n)) = A007814(n). Also a(A235489(n)) = a(n). - Antti Karttunen, Jan 27 2014

A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

1, 2, 3, 2, 4, 3, 3, 2, 3, 4, 5, 3, 4, 3, 4, 2, 4, 3, 3, 4, 4, 5, 4, 3, 4, 4, 3, 3, 5, 4, 6, 2, 5, 4, 5, 3, 4, 3, 4, 4, 5, 4, 4, 5, 4, 4, 5, 3, 3, 4, 5, 4, 3, 3, 5, 3, 4, 5, 5, 4, 4, 6, 4, 2, 5, 5, 4, 4, 4, 5, 5, 3, 5, 4, 4, 3, 6, 4, 6, 4, 3, 5, 5, 4, 6, 4, 5, 5, 4, 4, 5, 4, 6, 5, 5, 3, 5, 3, 5, 4, 5, 5, 4, 4, 5, 3, 4, 3

Offset: 1

Gus Wiseman, Aug 05 2018

nonn

			20 is the Matula-Goebel number of the tree (oo((o))), which has 4 distinct terminal subtrees: {(oo((o))), ((o)), (o), o}. So a(20) = 4.
See also illustrations in A061773.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to Matula-Göbel numbers

One more than A324923.

Cf. A000081, A007097, A049076, A061773, A061775, A076146, A109082, A109129, A206491, A303431, A316476.

ids[n_]:=Union@@FixedPointList[Union@@(Cases[If[#==1,{},FactorInteger[#]],{p_,_}:>PrimePi[p]]&/@#)&,{n}];
Table[Length[ids[n]],{n,100}]

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A324923(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #Set(lista); }; \\ Antti Karttunen, Oct 23 2023
A317713(n) = (1+A324923(n)); \\ Antti Karttunen, Oct 23 2023

a(n) = 1+A324923(n). - Antti Karttunen, Oct 23 2023

Data section extended up to a(108) by Antti Karttunen, Oct 23 2023

A227413 a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)), where nthprime = A000040, nthcomposite = A002808.

Original entry on oeis.org

1, 2, 4, 3, 6, 7, 9, 5, 8, 13, 12, 17, 14, 23, 16, 11, 10, 19, 15, 41, 22, 37, 21, 59, 27, 43, 24, 83, 35, 53, 26, 31, 20, 29, 18, 67, 30, 47, 25, 179, 58, 79, 34, 157, 54, 73, 33, 277, 82, 103, 40, 191, 62, 89, 36, 431, 114, 149, 51, 241, 75, 101, 39, 127, 46

Offset: 1

Antti Karttunen, Jul 10 2013

Inverse permutation of A135141.

Shares with A073846 the property that the other bisection consists of just primes and the other bisection of just nonprimes.

Antti Karttunen and Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Index entries for sequences that are permutations of the natural numbers

Similarly constructed permutations: A227402, A227404, A227410, A227412. Cf. also A073846, A209636.

import Data.List (transpose)
a227413 n = a227413_list !! (n-1)
a227413_list = 1 : concat (transpose [map a000040 a227413_list,
                                      map a002808 a227413_list])
-- Reinhard Zumkeller, Jan 29 2014

a(1)=1, a(2n) = A000040(a(n)), a(2n+1) = A002808(a(n)).

A007097(n) = a(A000079(n)).

A330944 Number of nonprime prime indices of n.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 1, 3, 0, 1, 0, 2, 1, 2, 0, 4, 0, 1, 1, 2, 1, 1, 1, 3, 0, 2, 0, 3, 1, 1, 0, 5, 0, 1, 1, 2, 1, 2, 1, 3, 0, 2, 1, 2, 0, 2, 1, 4, 2, 1, 0, 3, 1, 1, 0, 4, 1, 2, 0, 2, 1, 1, 1, 6, 1, 1, 0, 2, 1, 2, 1, 3, 1, 2, 0, 3, 1, 2, 1, 4, 0, 1, 0, 3, 0, 2, 1

Offset: 1

Gus Wiseman, Jan 13 2020

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.

			24 has prime indices {1,1,1,2}, of which {1,1,1} are nonprime, so a(24) = 3.

The number of prime prime indices is given by A257994.
Primes of prime index are A006450.
Primes of nonprime index are A007821.
Products of primes of prime index are A076610.
Products of primes of nonprime index are A320628.
Numbers whose prime indices are not all prime are A330945.
Cf. A000040, A000720, A001222, A007097, A018252, A056239, A112798, A302242, A320629, A320633, A330946, A330947.

Table[Total[Cases[If[n==1,{},FactorInteger[n]],{p_,k_}/;!PrimeQ[PrimePi[p]]:>k]],{n,30}]

a(n) = my(f=factor(n)); sum(k=1, #f~, if(!isprime(primepi(f[k,1])), f[k,2], 0)); \\ Daniel Suteu, Jan 14 2020

a(n) + A257994(n) = A001222(n).

Additive with a(p^e) = e if primepi(p) is nonprime, and 0 otherwise. - Amiram Eldar, Nov 03 2023

A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 16, 17, 19, 20, 21, 22, 23, 25, 27, 29, 31, 32, 33, 34, 35, 37, 40, 41, 43, 44, 46, 47, 49, 50, 51, 53, 57, 59, 61, 62, 63, 64, 65, 67, 68, 71, 73, 77, 79, 80, 81, 82, 83, 85, 87, 88, 89, 91, 92, 93, 94, 95, 97, 99, 100, 101

Offset: 1

Gus Wiseman, Mar 17 2019

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 Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
These could be described as anti-transitive numbers (cf. A290822), as they are numbers x such that if prime(y) divides x and prime(z) divides y, then prime(z) does not divide x.
Also numbers n such that A003963(n) is coprime to n.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   3: {2}
   4: {1,1}
   5: {3}
   7: {4}
   8: {1,1,1}
   9: {2,2}
  10: {1,3}
  11: {5}
  13: {6}
  16: {1,1,1,1}
  17: {7}
  19: {8}
  20: {1,1,3}
  21: {2,4}
  22: {1,5}
  23: {9}
  25: {3,3}
  27: {2,2,2}

The subset version is A324741, with maximal case A324743. The strict integer partition version is A324751. The integer partition version is A324756. An infinite version is A324695.

Cf. A000720, A001221, A001462, A007097, A056239, A112798, A276625, A289509, A290822, A304360, A306844, A324742, A324753, A324764.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Intersection[primeMS[#],Union@@primeMS/@primeMS[#]]=={}&]

A106349 Primes indexed by semiprimes.

Original entry on oeis.org

7, 13, 23, 29, 43, 47, 73, 79, 97, 101, 137, 139, 149, 163, 167, 199, 227, 233, 257, 269, 271, 293, 313, 347, 373, 389, 421, 439, 443, 449, 467, 487, 491, 499, 577, 607, 631, 647, 653, 661, 673, 677, 727, 751, 757, 811, 821, 823, 829, 839, 907, 929, 937, 947

Offset: 1

Jonathan Vos Post, Apr 29 2005

This is the sequence of the k-th prime for k = {4,6,9,10,14,15,21,22,25,26,33,34,35,38,39,46,49,51,...}. Not to be confused with A106350: semiprimes indexed by primes.

			a(1) = 7 because semiprime(1) = 4, so prime(semiprime(1)) = prime(4) = 7.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Paul Kinlaw, Megan Triplett, and William Tripp, Almost Primes of Almost Prime Index, INTEGERS, Vol 24 (2024), Article #A99.

Cf. A000040, A000720, A001358, A007097, A091022, A105997, A105998, A106350.

[NthPrime(n): n in [2..200] | &+[d[2]: d in Factorization(n)] eq 2]; // Vincenzo Librandi, Nov 28 2015

Prime@ Select[Range@ 161, PrimeOmega@ # == 2 &] (* or *) Select[Prime@ Range@ 161, PrimeOmega@ PrimePi@ # == 2 &] (* Michael De Vlieger, Nov 28 2015 *)

lista(nn) = select(x->(bigomega(primepi(x))==2), primes(nn)); \\ Michel Marcus, Nov 29 2015

a(n) = prime(semiprime(n)).
a(n) = A000040(A001358(n)).
pi(a(n)) = p*q for some primes p and q.
Sum_{n>=1} 1/a(n) is in the interval (0.9910, 0.9915) (Kinlaw et al., 2024, Theorem 6, p. 11). - Amiram Eldar, Nov 09 2024

A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 1, 3, 2, 2, 1, 3, 3, 4, 2, 3, 2, 4, 1, 3, 3, 2, 3, 3, 4, 4, 2, 5, 3, 4, 2, 4, 4, 5, 1, 5, 3, 4, 3, 3, 2, 4, 3, 4, 3, 3, 4, 5, 4, 5, 2, 3, 5, 4, 3, 2, 4, 6, 2, 3, 4, 4, 4, 4, 5, 4, 1, 5, 5, 3, 3, 5, 4, 4, 3, 4, 3, 6, 2, 5, 4, 5, 3, 5, 4, 5, 3, 5, 3, 5, 4, 3, 5, 4, 4, 6, 5, 4, 2, 6, 3, 6, 5

Offset: 1

François Marques, Mar 14 2021

nonn

The label f(T) for a rooted tree T is 1 if T has 1 node, otherwise f(T) = Product_{T_i} prime(f(T_i)) where the T_i are the subtrees obtained by deleting the root and the edges adjacent to it. (Cf. A061773 for illustration.)

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = 1 because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

François Marques, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Matula-Goebel numbers

Other statistics are: A061775 (nodes), A109082 (edge-height), A109129 (leaves), A196050 (edges), A358552 (node-height).
An ordered version is A358553.
Positions of first appearances are A358554.
A000081 counts rooted trees, ordered A000108.
A358575 counts rooted trees by nodes and internals.
Cf. A000040, A000720, A001222, A007097, A056239, A112798.
Cf. A034781, A055277, A206487, A358576, A358578, A358592.

MGTree[n_]:=If[n==1,{},MGTree/@Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Count[MGTree[n],[_],{0,Infinity}],{n,100}] (* Gus Wiseman, Nov 28 2022 *)

A342507(n) = if( n==1, 0, my(f=factor(n)); 1+sum(k=1,matsize(f)[1],A342507(primepi(f[k,1]))*f[k,2]));

a(1)=0 and a(n) = A061775(n) - A109129(n) for n > 1.

A109301 a(n) = rhig(n) = rote height in gammas of n, where the "rote" corresponding to a positive integer n is a graph derived from the primes factorization of n, as illustrated in the comments.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 3, 2, 3, 4, 2, 3, 3, 3, 3, 4, 2, 4, 3, 3, 4, 3, 3, 3, 3, 3, 3, 4, 3, 5, 4, 4, 4, 3, 2, 3, 4, 3, 3, 4, 3, 4, 4, 3, 3, 4, 3, 3, 3, 4, 3, 4, 3, 4, 3, 4, 4, 5, 3, 3, 5, 3, 3, 3, 4, 5, 4, 3, 3, 4, 3, 4, 3, 3, 4, 4, 3, 5, 3, 3, 4, 4, 3, 4, 4, 4, 4, 4, 3, 3, 3, 5, 4, 4, 4, 4, 3, 4, 3

Offset: 1

Jon Awbrey, Jun 24 2005 - Jul 08 2005

nonn

Table of Rotes and Primal Functions for Positive Integers from 1 to 40
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` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` o-o ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` `
` ` ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` `
` ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` `
O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
{ } ` ` ` ` ` 1:1 ` ` ` ` ` 2:1 ` ` ` ` ` 1:2 ` ` ` ` ` 3:1 ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1 ` ` ` ` ` ` 2 ` ` ` ` ` ` 3 ` ` ` ` ` ` 4 ` ` ` ` ` ` 5 ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` `
` ` o-o ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` o-o o-o ` ` ` ` ` o-o ` ` ` `
` ` | ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` `
o-o o-o ` ` ` o-o ` ` ` ` ` o-o ` ` ` ` ` o---o ` ` ` ` o-o o-o ` ` ` `
| ` | ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` `
O===O ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1:1 2:1 ` ` ` 4:1 ` ` ` ` ` 1:3 ` ` ` ` ` 2:2 ` ` ` ` ` 1:1 3:1 ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
6 ` ` ` ` ` ` 7 ` ` ` ` ` ` 8 ` ` ` ` ` ` 9 ` ` ` ` ` ` 10` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` o-o ` ` ` `
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` `
o-o ` ` ` ` ` ` o-o o-o ` ` o-o o-o ` ` ` ` ` o-o ` ` ` o-o o-o ` ` ` `
| ` ` ` ` ` ` ` | ` | ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` | ` | ` ` ` ` `
o-o ` ` ` ` ` o-o ` o-o ` ` o===o-o ` ` ` o-o o-o ` ` ` o-o o-o ` ` ` `
| ` ` ` ` ` ` | ` ` | ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` `
O ` ` ` ` ` ` O=====O ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
5:1 ` ` ` ` ` 1:2 2:1 ` ` ` 6:1 ` ` ` ` ` 1:1 4:1 ` ` ` 2:1 3:1 ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
11` ` ` ` ` ` 12` ` ` ` ` ` 13` ` ` ` ` ` 14` ` ` ` ` ` 15` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` `
` ` o-o ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` o-o ` ` `
` ` | ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` `
` o-o ` ` ` ` o-o ` ` ` ` ` ` ` o-o o-o ` o-o ` ` ` ` ` ` o-o o-o ` ` `
` | ` ` ` ` ` | ` ` ` ` ` ` ` ` | ` | ` ` | ` ` ` ` ` ` ` | ` | ` ` ` `
o-o ` ` ` ` ` o-o ` ` ` ` ` o-o o---o ` ` o-o ` ` ` ` ` o-o ` o-o ` ` `
| ` ` ` ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` ` ` ` ` ` | ` ` | ` ` ` `
O ` ` ` ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O ` ` ` ` ` ` O=====O ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1:4 ` ` ` ` ` 7:1 ` ` ` ` ` 1:1 2:2 ` ` ` 8:1 ` ` ` ` ` 1:2 3:1 ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
16` ` ` ` ` ` 17` ` ` ` ` ` 18` ` ` ` ` ` 19` ` ` ` ` ` 20` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
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` ` ` o-o ` ` ` ` o-o ` ` ` o-o o-o ` ` ` ` o-o ` ` ` ` o-o ` ` ` ` ` `
` ` ` | ` ` ` ` ` | ` ` ` ` | ` | ` ` ` ` ` | ` ` ` ` ` | ` ` ` ` ` ` `
o-o o-o ` ` ` ` ` o-o ` ` ` o---o ` ` ` ` ` o-o o-o ` ` o-o o-o ` ` ` `
| ` | ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` | ` | ` ` ` | ` | ` ` ` ` `
o-o o-o ` ` ` o-o o-o ` ` ` o-o ` ` ` ` ` o-o ` o-o ` ` o---o ` ` ` ` `
| ` | ` ` ` ` | ` | ` ` ` ` | ` ` ` ` ` ` | ` ` | ` ` ` | ` ` ` ` ` ` `
O===O ` ` ` ` O===O ` ` ` ` O ` ` ` ` ` ` O=====O ` ` ` O ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
2:1 4:1 ` ` ` 1:1 5:1 ` ` ` 9:1 ` ` ` ` ` 1:3 2:1 ` ` ` 3:2 ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
21` ` ` ` ` ` 22` ` ` ` ` ` 23` ` ` ` ` ` 24` ` ` ` ` ` 25` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` o-o ` ` ` o-o ` ` ` ` ` ` ` o-o ` ` ` o-o ` ` ` ` ` ` ` o-o ` `
` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` ` | ` ` `
` ` o-o o-o ` o-o o-o ` ` ` ` o-o o-o ` ` o-o o-o ` ` ` ` ` o-o o-o ` `
` ` | ` | ` ` | ` | ` ` ` ` ` | ` | ` ` ` | ` | ` ` ` ` ` ` | ` | ` ` `
o-o o===o-o ` o---o ` ` ` ` o-o ` o-o ` ` o===o-o ` ` ` o-o o-o o-o ` `
| ` | ` ` ` ` | ` ` ` ` ` ` | ` ` | ` ` ` | ` ` ` ` ` ` | ` | ` | ` ` `
O===O ` ` ` ` O ` ` ` ` ` ` O=====O ` ` ` O ` ` ` ` ` ` O===O===O ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1:1 6:1 ` ` ` 2:3 ` ` ` ` ` 1:2 4:1 ` ` ` 10:1` ` ` ` ` 1:1 2:1 3:1 ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
26` ` ` ` ` ` 27` ` ` ` ` ` 28` ` ` ` ` ` 29` ` ` ` ` ` 30` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` `
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` `
o-o ` ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` ` ` ` o-o ` ` ` o-o ` o-o ` ` `
| ` ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` ` ` ` | ` ` ` ` | ` ` | ` ` ` `
o-o ` ` ` ` ` ` o-o ` ` ` ` o-o o-o ` ` ` ` ` o-o ` ` ` o-o o-o ` ` ` `
| ` ` ` ` ` ` ` | ` ` ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` | ` | ` ` ` ` `
o-o ` ` ` ` ` o-o ` ` ` ` ` o-o o-o ` ` ` o-o o-o ` ` ` o-o o-o ` ` ` `
| ` ` ` ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` `
O ` ` ` ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
11:1` ` ` ` ` 1:5 ` ` ` ` ` 2:1 5:1 ` ` ` 1:1 7:1 ` ` ` 3:1 4:1 ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
31` ` ` ` ` ` 32` ` ` ` ` ` 33` ` ` ` ` ` 34` ` ` ` ` ` 35` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` o-o ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` o-o o-o ` ` ` ` ` o-o ` ` ` ` ` ` o-o ` ` o-o o-o ` ` `
` ` ` ` ` ` ` ` | ` | ` ` ` ` ` ` | ` ` ` ` ` ` ` | ` ` ` | ` | ` ` ` `
` o-o o-o o-o o-o ` o-o ` ` ` ` o-o ` ` ` o-o o-o o-o ` ` o-o o-o ` ` `
` | ` | ` | ` | ` ` | ` ` ` ` ` | ` ` ` ` | ` | ` | ` ` ` | ` | ` ` ` `
o-o ` o---o ` o=====o-o ` ` o-o o-o ` ` ` o-o o===o-o ` o-o ` o-o ` ` `
| ` ` | ` ` ` | ` ` ` ` ` ` | ` | ` ` ` ` | ` | ` ` ` ` | ` ` | ` ` ` `
O=====O ` ` ` O ` ` ` ` ` ` O===O ` ` ` ` O===O ` ` ` ` O=====O ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
1:2 2:2 ` ` ` 12:1` ` ` ` ` 1:1 8:1 ` ` ` 2:1 6:1 ` ` ` 1:3 3:1 ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
36` ` ` ` ` ` 37` ` ` ` ` ` 38` ` ` ` ` ` 39` ` ` ` ` ` 40` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `
In these Figures, "extended lines of identity" like o===o indicate identified nodes and capital O is the root node. The rote height in gammas is found by finding the number of graphs of the following shape between the root and one of the highest nodes of the tree:
o--o
|
o
A sequence like this, which can be regarded as a nonnegative integer measure on positive integers, may have as many as 3 other sequences associated with it. Given that the fiber of a function f at n is all the domain elements that map to n, we always have the fiber minimum or minimum inverse function and may also have the fiber cardinality and the fiber maximum or maximum inverse function. For A109301, the minimum inverse is A007097(n) = min {k : A109301(k) = n}, giving the first positive integer whose rote height is n; the fiber cardinality is A109300, giving the number of positive integers of rote height n; the maximum inverse, g(n) = max {k : A109301(k) = n}, giving the last positive integer whose rote height is n, has the following initial terms: g(0) = { } = 1, g(1) = 1:1 = 2, g(2) = 1:2 2:2 = 36, while g(3) = 1:36 2:36 3:36 4:36 6:36 9:36 12:36 18:36 36:36 = (2 3 5 7 13 23 37 61 151)^36 = 21399271530^36 = roughly 7.840858554516122655953405327738 x 10^371.

			Writing (prime(i))^j as i:j, we have:
802701 = 2:2 8638:1
8638 = 1:1 4:1 113:1
113 = 30:1
30 = 1:1 2:1 3:1
4 = 1:2
3 = 2:1
2 = 1:1
1 = { }
So rote(802701) is the graph:
` ` ` ` ` ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` o-o
` ` ` ` ` ` ` ` ` ` ` ` ` | `
` ` ` ` ` ` ` ` ` ` ` o-o o-o
` ` ` ` ` ` ` ` ` ` ` | ` | `
` ` ` ` ` ` ` o-o o-o o-o o-o
` ` ` ` ` ` ` | ` | ` | ` | `
` ` ` ` ` ` o-o ` o===o===o-o
` ` ` ` ` ` | ` ` | ` ` ` ` `
o-o o-o o-o o-o ` o---------o
| ` | ` | ` | ` ` | ` ` ` ` `
o---o ` o===o=====o---------o
| ` ` ` | ` ` ` ` ` ` ` ` ` `
O=======O ` ` ` ` ` ` ` ` ` `
` ` ` ` ` ` ` ` ` ` ` ` ` ` `
Therefore rhig(802701) = 6.

J. Awbrey, Riffs and Rotes

Cf. A007097, A050924, A061396, A062504, A062537, A062860.

Cf. A106177, A108352, A108371, A109300, A111791 to A111800.

Writing (prime(i))^j as i:j, the prime factorization of a positive integer n can be written as n = prod_(k = 1 to m) i(k):j(k). This sets up the formula: rhig(n) = 1 + max_(k = 1 to m) {rhig(i(k)), rhig(j(k))}, where rhig(1) = 0.

A290760 Matula-Goebel numbers of transitive rooted identity trees (or transitive finitary sets).

Original entry on oeis.org

1, 2, 6, 30, 78, 330, 390, 870, 1410, 3198, 3390, 4290, 7878, 9570, 10230, 11310, 13026, 15510, 15990, 18330, 26070, 30966, 37290, 39390, 40890, 44070, 45210, 65130, 84810, 94830, 98310, 104610, 122070, 124410, 132990, 154830, 159330, 175890, 198330, 201630

Offset: 1

Gus Wiseman, Oct 19 2017

nonn

A rooted tree is transitive if every terminal subtree is a branch of the root. A finitary set is transitive if every element is also a subset.

			Let o = {}. The sequence of transitive finitary sets begins:
1     o
2     {o}
6     {o,{o}}
30    {o,{o},{{o}}}
78    {o,{o},{o,{o}}}
330   {o,{o},{{o}},{{{o}}}}
390   {o,{o},{{o}},{o,{o}}}
870   {o,{o},{{o}},{o,{{o}}}}
1410  {o,{o},{{o}},{{o},{{o}}}}
3198  {o,{o},{o,{o}},{{o,{o}}}}
3390  {o,{o},{{o}},{o,{o},{{o}}}}
4290  {o,{o},{{o}},{{{o}}},{o,{o}}}
7878  {o,{o},{o,{o}},{o,{o,{o}}}}
9570  {o,{o},{{o}},{{{o}}},{o,{{o}}}}
10230 {o,{o},{{o}},{{{o}}},{{{{o}}}}}
11310 {o,{o},{{o}},{o,{o}},{o,{{o}}}}
13026 {o,{o},{o,{o}},{{o},{o,{o}}}}
15510 {o,{o},{{o}},{{{o}}},{{o},{{o}}}}
15990 {o,{o},{{o}},{o,{o}},{{o,{o}}}}
18330 {o,{o},{{o}},{o,{o}},{{o},{{o}}}}

Cf. A000081, A001192, A004111, A007097, A076146, A276625, A279861, A290689, A290822.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
finitaryQ[n_]:=finitaryQ[n]=Or[n===1,With[{m=primeMS[n]},{UnsameQ@@m,finitaryQ/@m}]/.List->And];
subprimes[n_]:=If[n===1,{},Union@@Cases[FactorInteger[n],{p_,_}:>FactorInteger[PrimePi[p]][[All,1]]]];
transitaryQ[n_]:=Divisible[n,Times@@subprimes[n]];
nn=100000;Fold[Select,Range[nn],{finitaryQ,transitaryQ}]

A127301 Matula-Goebel signatures for plane general trees encoded by A014486.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 6, 7, 5, 16, 12, 12, 14, 10, 12, 9, 14, 19, 13, 10, 13, 17, 11, 32, 24, 24, 28, 20, 24, 18, 28, 38, 26, 20, 26, 34, 22, 24, 18, 18, 21, 15, 28, 21, 38, 53, 37, 26, 37, 43, 29, 20, 15, 26, 37, 23, 34, 43, 67, 41, 22, 29, 41, 59, 31, 64, 48, 48, 56, 40, 48, 36

Offset: 0

Antti Karttunen, Jan 16 2007

nonn

This sequence maps A000108(n) oriented (plane) rooted general trees encoded in range [A014137(n-1)..A014138(n)] of A014486 to A000081(n+1) distinct non-oriented rooted general trees, encoded by their Matula-Goebel numbers. The latter encoding is explained in A061773.
A005517 and A005518 give the minimum and maximum value occurring in each such range.
Primes occur at positions given by A057548 (not in order, and with duplicates), and similarly, semiprimes, A001358, occur at positions given by A057518, and in general, A001222(a(n)) = A057515(n).
If the signature-permutation of a Catalan automorphism SP satisfies the condition A127301(SP(n)) = A127301(n) for all n, then it preserves the non-oriented form of a general tree, which implies also that it is Łukasiewicz-word permuting, satisfying A129593(SP(n)) = A129593(n) for all n >= 0. Examples of such automorphisms include A072796, A057508, A057509/A057510, A057511/A057512, A057164, A127285/A127286 and A127287/A127288.
A206487(n) tells how many times n occurs in this sequence. - Antti Karttunen, Jan 03 2013

			A000081(n+1) distinct values occur each range [A014137(n-1)..A014138(n-1)]. As an example, A014486(5) = 44 (= 101100 in binary = A063171(5)), encodes the following plane tree:
.....o
.....|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(1) * A000040(A000040(1)) = 2*3 = 6, thus a(5)=6.
Likewise, A014486(6) = 50 (= 110010 in binary = A063171(6)) encodes the plane tree:
.o
.|
.o...o
..\./.
...*..
Matula-Goebel encoding for this tree gives a code number A000040(A000040(1)) * A000040(1) = 3*2 = 6, thus a(6) is also 6, which shows these two trees are identical if one ignores their orientation.

Antti Karttunen, Table of n, a(n) for n = 0..6917
OEIS Wiki, Łukasiewicz words
Index entries for sequences related to Łukasiewicz
Index entries for sequences related to Matula-Goebel numbers

a(A014138(n)) = A007097(n+1), a(A014137(n)) = A000079(n+1) for all n.
a(|A106191(n)|) = A033844(n-1) for all n >= 1.
Cf. A001222, A005517, A005518, A057515, A057518, A057548, A127302, A129593, A153826, A209638, A243491, A243492, A243494, A243496.
For standard instead of binary encoding we have A358506.
A000108 counts ordered rooted trees, unordered A000081.
A014486 lists binary encodings of ordered rooted trees.
Cf. A001263, A061775, A196050, A206487, A358505.

mgnum[t_]:=If[t=={},1,Times@@Prime/@mgnum/@t];
binbalQ[n_]:=n==0||With[{dig=IntegerDigits[n,2]},And@@Table[If[k==Length[dig],SameQ,LessEqual][Count[Take[dig,k],0],Count[Take[dig,k],1]],{k,Length[dig]}]];
bint[n_]:=If[n==0,{},ToExpression[StringReplace[StringReplace[ToString[IntegerDigits[n,2]/.{1->"{",0->"}"}],","->""],"} {"->"},{"]]];
Table[mgnum[bint[n]],{n,Select[Range[0,1000],binbalQ]}] (* Gus Wiseman, Nov 22 2022 *)

(define (A127301 n) (*A127301 (A014486->parenthesization (A014486 n)))) ;; A014486->parenthesization given in A014486.
(define (*A127301 s) (if (null? s) 1 (fold-left (lambda (m t) (* m (A000040 (*A127301 t)))) 1 s)))

A001222(a(n)) = A057515(n) for all n.

cp's OEIS Frontend

A078442 a(p) = a(n) + 1 if p is the n-th prime, prime(n); a(n)=0 if n is not prime.

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A317713 Number of distinct terminal subtrees of the rooted tree with Matula-Goebel number n.

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A227413 a(1)=1, a(2n)=nthprime(a(n)), a(2n+1)=nthcomposite(a(n)), where nthprime = A000040, nthcomposite = A002808.

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A330944 Number of nonprime prime indices of n.

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A324758 Heinz numbers of integer partitions containing no prime indices of the parts.

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A106349 Primes indexed by semiprimes.

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A342507 Number of internal nodes in rooted tree with Matula-Goebel number n.

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