A324923 -id:A324923 - cp's OEIS frontend

A366386 Excess of n when n is factored into factors q(i) = prime(i)/i: a(n) = A196050(n) - A324923(n).

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

Offset: 1

Antti Karttunen, Oct 23 2023

nonn

Cf. A006530, A196050, A324923.
Cf. A007097 (positions of 0's), A358731 (of 1's).
Cf. also A046660, A358729.

A006530(n) = if(1==n, n, my(f=factor(n)); f[#f~, 1]);
A366386(n) = { my(lista = List([]), gpf, i); while(n > 1, gpf=A006530(n); i = primepi(gpf); n /= gpf; n *= i; listput(lista,i)); #lista - #Set(lista); };

a(n) = A196050(n) - A324923(n).

a(n) <= A358729(n).

A196050 Number of edges in the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

0, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 4, 5, 4, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 6, 5, 6, 6, 5, 5, 6, 6, 5, 6, 5, 6, 7, 6, 6, 6, 6, 7, 6, 6, 5, 7, 7, 6, 6, 6, 5, 7, 6, 6, 7, 6, 7, 7, 5, 6, 7, 7, 6, 7, 6, 6, 8, 6, 7, 7, 6, 7, 8, 6, 6, 7, 7, 6, 7, 7, 6, 8, 7, 7, 7, 7, 7, 7, 7, 7, 8, 8, 6, 7, 7, 7, 8, 6, 6, 8, 6, 8

Offset: 1

Emeric Deutsch, Sep 27 2011

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
a(n) is, for n >= 2, the number of prime function prime(.) = A000040(.) operations in the complete reduction of n. See the W. Lang link with a list of the reductions for n = 2..100, where a curly bracket notation {.} is used for prime(.). - Wolfdieter Lang, Apr 03 2018
From Gus Wiseman, Mar 23 2019: (Start)
Every positive integer has a unique factorization (encoded by A324924) into factors q(i) = prime(i)/i, i > 0. For example:
   11 = q(1) q(2) q(3) q(5)
   50 = q(1)^3 q(2)^2 q(3)^2
  360 = q(1)^6 q(2)^3 q(3)
In this factorization, a(n) is the number of factors counted with multiplicity. For example, a(11) = 4, a(50) = 7, a(360) = 10.
(End)
From Antti Karttunen, Oct 23 2023: (Start)
Totally additive with a(prime(n)) = 1 + a(n).
Number of iterations of A366385 (or equally, of A366387) needed to reach 1.
(End)

			a(7) = 3 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y.
a(2^m) = m because the rooted tree with Matula-Goebel number 2^m is the star tree with m edges.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Emeric Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Göbel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
Wolfdieter Lang, Complete prime function reduction for n = 2..100.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

One less than A061775.
Cf. A000040, A000081, A000720, A003963, A007097, A020639, A049084, A109082, A109129, A317713, A318995, A358729.
Cf. A324850, A324922, A324923, A324924, A324925, A324931, A324935, A366385, A366387, A366388.

import Data.List (genericIndex)
a196050 n = genericIndex a196050_list (n - 1)
a196050_list = 0 : g 2 where
   g x = y : g (x + 1) where
     y = if t > 0 then a196050 t + 1 else a196050 r + a196050 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif bigomega(n) = 1 then 1+a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

a[1] = 0; a[n_?PrimeQ] := a[n] = 1 + a[PrimePi[n]]; a[n_] := Total[#[[2]] * a[#[[1]] ]& /@ FactorInteger[n]];
Array[a, 110] (* Jean-François Alcover, Nov 16 2017 *)
difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[Length[difac[n]],{n,100}] (* Gus Wiseman, Mar 23 2019 *)

a(n) = my(f=factor(n)); [self()(primepi(p))+1 |p<-f[,1]]*f[,2]; \\ Kevin Ryde, May 28 2021

from functools import lru_cache
from sympy import isprime, primepi, factorint
@lru_cache(maxsize=None)
def A196050(n):
    if n == 1 : return 0
    if isprime(n): return 1+A196050(primepi(n))
    return sum(e*A196050(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1)=0; if n = prime(t) (the t-th prime), then a(n)=1 + a(t); if n = r*s (r,s>=2), then a(n)=a(r)+a(s). The Maple program is based on this recursive formula.
a(n) = A061775(n) - 1.
a(n) = A109129(n) + A366388(n) = A109082(n) + A358729(n). - Antti Karttunen, Oct 23 2023

A109129 Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n.

Original entry on oeis.org

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

Offset: 1

Keith Briggs, Aug 17 2005

nonn

The Matula-Goebel number of a rooted tree is defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m >= 2 there corresponds the product of the Matula-Goebel numbers of the m branches of T.
A non-root vertex having degree 1 is called a leaf.
Every positive integer has a unique factorization (see A324924) into factors q(i) = prime(i)/i for i > 0. The number of ones in this factorization is a(n). For example, 30 = q(1)^3 q(2)^2 q(3), so a(30) = 3. - Gus Wiseman, Mar 23 2019

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.
F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
D. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

Cf. A061775, A091233.
Cf. A049084, A020639.
Cf. A000081, A000720, A001222, A007097, A109082, A196050, A317713.
Cf. A324850, A324922, A324923, A324924, A324931.

import Data.List (genericIndex)
a109129 n = genericIndex a109129_list (n - 1)
a109129_list = 0 : 1 : g 3 where
   g x = y : g (x + 1) where
     y = if t > 0 then a109129 t else a109129 r + a109129 s
         where t = a049084 x; r = a020639 x; s = x `div` r
-- Reinhard Zumkeller, Sep 03 2013

with(numtheory): a := proc (n) local r, s: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n)) else a(r(n))+a(s(n)) end if end proc: seq(a(n), n = 1 .. 110);

Nest[Function[{a, n}, Append[a, If[PrimeQ@ n, a[[PrimePi@ n]], Total@ Map[#2 a[[#1]] & @@ # &, FactorInteger[n]] ]]] @@ {#, Length@ # + 1} &, {0, 1}, 105] (* Michael De Vlieger, Mar 24 2019 *)

ML(n) = if( n==1, 1, my(f=factor(n)); sum(k=1,matsize(f)[1],ML(primepi(f[k,1]))*f[k,2])) ;
A109129(n) = if( n==1, 0, ML(n) ); \\ François Marques, Mar 16 2021

from functools import lru_cache
from sympy import primepi, isprime, factorint
@lru_cache(maxsize=None)
def A109129(n):
    if n <= 2: return n-1
    if isprime(n): return A109129(primepi(n))
    return sum(e*A109129(p) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 19 2022

a(1)=0; a(2)=1; if n = p(t) (= the t-th prime) and t >= 2, then a(n) = a(t); if n = rs (r, s >= 2), then a(n) = a(r) + a(s). The Maple program is based on this recursive formula.

The Gutman et al. references contain a different recursive formula.

Typo in formula fixed by Reinhard Zumkeller, Sep 03 2013

A324850 Numbers divisible by the product of their prime indices.

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 24, 28, 30, 32, 36, 48, 56, 60, 64, 72, 96, 112, 120, 128, 144, 152, 156, 168, 180, 192, 216, 224, 240, 256, 288, 304, 312, 330, 336, 360, 384, 432, 448, 476, 480, 512, 576, 608, 624, 660, 672, 720, 768, 784, 828, 840, 848, 864, 888, 896

Offset: 1

Gus Wiseman, Mar 18 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, with product A003963(n). For example, the prime indices of 30 are {1,2,3}, with product 6, which divides 30, so 30 is in the sequence.

			The sequence of terms together with their prime indices begins:
   1: {}
   2: {1}
   4: {1,1}
   6: {1,2}
   8: {1,1,1}
  12: {1,1,2}
  16: {1,1,1,1}
  24: {1,1,1,2}
  28: {1,1,4}
  30: {1,2,3}
  32: {1,1,1,1,1}
  36: {1,1,2,2}
  48: {1,1,1,1,2}
  56: {1,1,1,4}
  60: {1,1,2,3}
  64: {1,1,1,1,1,1}
  72: {1,1,1,2,2}
  96: {1,1,1,1,1,2}

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A003963, A036844, A120383, A324847, A324756, A324758, A324847, A324848, A324849, A324852, A324853, A324856, A324923, A324924, A324925, A324931.

Select[Range[100],Divisible[#,Times@@Cases[If[#==1,{},FactorInteger[#]],{p_,k_}:>PrimePi[p]^k]]&]

isok(n) = my(f=factor(n)); !(n % prod(k=1, #f~, primepi(f[k,1])^f[k,2])); \\ Michel Marcus, Mar 22 2019

n/A003963(n) = A324933(n)/A324934(n).

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

A324924 Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 4, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 3, 1, 2, 3, 5, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 4, 1, 1, 2, 2, 3, 1, 1, 1, 1, 1, 1, 4, 7, 1, 1, 1, 2, 2, 1, 1, 1, 8, 1, 1, 1, 2, 3, 1, 1, 1, 2, 4, 1, 1, 2, 3, 5, 1, 1, 2, 2, 9

Offset: 1

Gus Wiseman, Mar 20 2019

Row n is the multiset of Matula-Goebel numbers of all proper terminal subtrees of the rooted tree with Matula-Goebel number n. For example, the rooted tree with Matula-Goebel number 1362 is (o(o)((oo)(oo))), with proper terminal subtrees {o,o,o,o,o,o,(o),(oo),(oo),((oo)(oo))}, which have Matula-Goebel numbers {1,1,1,1,1,1,2,4,4,49}, which is row 1362, as required.

			Triangle begins:
  {}
  1
  1  2
  1  1
  1  2  3
  1  1  2
  1  1  4
  1  1  1
  1  1  2  2
  1  1  2  3
  1  2  3  5
  1  1  1  2
  1  1  2  6
  1  1  1  4
  1  1  2  2  3
  1  1  1  1
  1  1  4  7
  1  1  1  2  2
  1  1  1  8
  1  1  1  2  3
  1  1  1  2  4
  1  1  2  3  5
  1  1  2  2  9
For example, row 65 is {1,1,1,2,2,3,6} because 65 = q(1)^3 q(2)^2 q(3) q(6).

Cf. A000081, A003963, A061775, A109082, A109129, A112798, A120383, A196050, A317713, A324850, A324922, A324923, A324925, A324931, A324934.

difac[n_]:=If[n==1,{},With[{i=PrimePi[FactorInteger[n][[1,1]]]},Sort[Prepend[difac[n*i/Prime[i]],i]]]];
Table[difac[n],{n,30}]

A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 28, 8, 36, 60, 330, 24, 156, 56, 180, 16, 476, 72, 152, 120, 168, 660, 828, 48, 900, 312, 216, 112, 1740, 360, 10230, 32, 1980, 952, 840, 144, 888, 304, 936, 240, 6396, 336, 2408, 1320, 1080, 1656, 8460, 96, 784, 1800, 2856, 624, 848, 432

Offset: 1

Gus Wiseman, Mar 20 2019

Every positive integer has a unique factorization into factors q(i) = prime(i)/i, i > 0 given by the rows of A324924. Then a(n) is the number obtained by encoding this factorization as a standard factorization into prime numbers (A112798).

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Sorting the sequence gives A324850.
Cf. A000081, A003963, A061775, A109129, A112798, A120383, A196050, A317713.
Cf. A324848, A324849, A324923, A324924, A324925, A324931, A324934.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
difac[n_]:=If[n==1,{},With[{m=Product[Prime[i]/i,{i,primeMS[n]}]},Sort[Join[primeMS[n],difac[n/m]]]]];
Table[Times@@Prime/@difac[n],{n,30}]

a(n) = my (f=factor(n)); prod (i=1, #f~, (f[i,1] * a(primepi(f[i,1])))^f[i,2]) \\ Rémy Sigrist, Jul 18 2019

a(n) = Product_t mg(t) where the product is over all (not necessarily distinct) terminal subtrees of the rooted tree with Matula-Goebel number n, and mg(t) is the Matula-Goebel number of t.

Completely multiplicative with a(prime(n)) = prime(n) * a(n). - Rémy Sigrist, Jul 18 2019

Keyword mult added by Rémy Sigrist, Jul 18 2019

A324931 Integers in the list of quotients of positive integers by their product of prime indices.

Original entry on oeis.org

1, 2, 4, 3, 8, 6, 16, 12, 7, 5, 32, 9, 24, 14, 10, 64, 18, 48, 28, 20, 128, 36, 19, 13, 21, 15, 96, 27, 56, 40, 256, 72, 38, 26, 11, 42, 30, 192, 54, 112, 17, 80, 512, 144, 76, 52, 22, 84, 60, 384, 49, 23, 35, 53, 108, 37, 224, 25, 57, 39, 34, 160, 63, 1024

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

These quotients are given by A324932(n)/A324933(n).

This is a permutation of the positive integers, with inverse A324934.

			The sequence of quotients n/A003963(n) begins: 1, 2, 3/2, 4, 5/3, 3, 7/4, 8, 9/4, 10/3, 11/5, 6, 13/6, 7/2, 5/2, 16, ...

Cf. A003963, A109129, A120383, A196050, A324846, A324849, A324850, A324922, A324923, A324924, A324925, A324934.

Select[Table[n/Times@@Cases[If[n==1,{},FactorInteger[n]],{p_,k_}:>PrimePi[p]^k],{n,100}],IntegerQ]

a(n) = A324850(n)/A003963(A324850(n)).

A324925 Number of integer partitions y of n such that Product_{i in y} prime(i)/i is an integer.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 5, 5, 5, 8, 9, 11, 17, 19, 21, 28, 32, 40, 51, 57, 67, 83, 96, 118, 142, 160, 189, 224, 260, 307, 363, 412, 479, 561, 649, 749, 874, 997, 1141, 1321, 1518, 1734, 1994, 2274, 2582, 2960, 3374, 3837, 4370, 4950, 5604, 6371, 7208, 8157, 9231, 10392

Offset: 0

Gus Wiseman, Mar 20 2019

nonn

The Heinz numbers of these integer partitions are given by A324850.

			The a(1) = 1 through a(8) = 5 integer partitions:
  (1)  (11)  (21)   (211)   (2111)   (321)     (3211)     (32111)
             (111)  (1111)  (11111)  (411)     (4111)     (41111)
                                     (2211)    (22111)    (221111)
                                     (21111)   (211111)   (2111111)
                                     (111111)  (1111111)  (11111111)
For example, (3,2,1,1) is such a partition because (2/1) * (2/1) * (3/2) * (5/3) = 10 is an integer.

Cf. A003963, A109129, A324850, A324922, A324923, A324924, A324931, A324934.

Table[Length[Select[IntegerPartitions[n],IntegerQ[Product[Prime[i]/i,{i,#}]]&]],{n,0,30}]

A324969 Number of unlabeled rooted identity trees with n vertices whose non-leaf terminal subtrees are all different.

Original entry on oeis.org

1, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, 4181, 6765, 10946, 17711, 28657, 46368, 75025, 121393, 196418, 317811, 514229, 832040, 1346269, 2178309, 3524578, 5702887, 9227465, 14930352, 24157817, 39088169, 63245986, 102334155

Offset: 1

Gus Wiseman, Mar 21 2019

A rooted identity tree is an unlabeled rooted tree with no repeated branches directly under the same root. This sequence counts rooted identity trees satisfying the additional condition that all non-leaf terminal subtrees are different.
Appears to be essentially the same as the Fibonacci sequence A000045. - R. J. Mathar, Mar 28 2019
From Michael Somos, Nov 22 2019: (Start)
A terminal subtree T' of a tree T is a subtree all of whose vertices except one have the same degree in T' as in T itself.
The conjecture of Mathar is true. Proof: Given a rooted identity tree T, a terminal subtree T' with more than one vertex contains at least one edge that is also a terminal subtree of T'. Thus, if T has more than one branch with more than one vertex, then it fails the additional condition since it would have at least two non-leaf terminal subtrees (namely edges) that are the same.  Also, T can't have under its root more than one branch with exactly one vertex because it is an identity tree. Now we know that under the root of T is exactly one branch of the same kind as T or else it has exactly one other branch with exactly one vertex. The leads immediately to the same recurrence as A000045 the Fibonacci sequence except for n=3. (End)

			The a(1) = 1 through a(7) = 8 trees:
  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))))))
G.f. = x + x^2 + x^3 + 2*x^4 + 3*x^5 + 5*x^6 + 8*x^7 + 13*x^8 + ... - _Michael Somos_, Nov 22 2019

G. C. Greubel, Table of n, a(n) for n = 1..1000
Joshua P. Bowman, Compositions with an Odd Number of Parts, and Other Congruences, J. Int. Seq (2024) Vol. 27, Art. 24.3.6. See p. 21.
Jia Huang, Partially Palindromic Compositions, J. Int. Seq. (2023) Vol. 26, Art. 23.4.1. See pp. 4, 11.
Index entries for linear recurrences with constant coefficients, signature (1,1).

The Matula-Goebel numbers of these trees are given by A324968.
Cf. A000045, A000081, A004111, A276625, A290689.
Cf. A317713, A324923, A324936, A324971, A324978.

[1] cat [Fibonacci(n-1): n in [2..50]]; // G. C. Greubel, Oct 24 2023

(* First program *)
durtid[n_]:= Join@@Table[Select[Union[Sort/@Tuples[durtid/@ptn]], UnsameQ@@#&&UnsameQ@@Cases[#, {}, {0,Infinity}]&],{ptn, IntegerPartitions[n-1]}];
Table[Length[durtid[n]],{n,15}]
(* Second program *)
Join[{1}, Fibonacci[Range[50]]] (* G. C. Greubel, Oct 24 2023 *)

{a(n) = if( n<=1, n==1, fibonacci(n-1))}; /* Michael Somos, Nov 22 2019 */

[int(n==1) +fibonacci(n-1) for n in range(1,51)] # G. C. Greubel, Oct 24 2023

From Michael Somos, Nov 22 2019: (Start)
G.f.: x*(1 - x^2) / (1 - x - x^2) = x*(1 + x/(1 - x/(1 - x/(1 + x)))).
a(n) = A000045(n-1) if n>=2. (End)
E.g.f.: -1 + x + exp(x/2)*(cosh(sqrt(5)*x/2) - (1/sqrt(5))*sinh(sqrt(5)*x/2)). - G. C. Greubel, Oct 24 2023

More terms from Jinyuan Wang, Jun 27 2020

cp's OEIS Frontend

A366386 Excess of n when n is factored into factors q(i) = prime(i)/i: a(n) = A196050(n) - A324923(n).

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A196050 Number of edges in the rooted tree with Matula-Goebel number n.

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A109129 Width (i.e., number of non-root vertices having degree 1) of the rooted tree with Matula-Goebel number n.

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A324850 Numbers divisible by the product of their prime indices.

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

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A324924 Irregular triangle read by rows giving the factorization of n into factors q(i) = prime(i)/i, i > 0.

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A324922 a(n) = unique m such that m/A003963(m) = n, where A003963 is product of prime indices.

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