A324928 -id:A324928 - cp's OEIS frontend

A109082 Depth of rooted tree having Matula-Goebel number n.

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

Offset: 1

Keith Briggs, Aug 17 2005

nonn

Another term for depth is height.

Starting with n, a(n) is the number of times one must take the product of prime indices (A003963) to reach 1. - Gus Wiseman, Mar 27 2019

			a(7) = 2 because the rooted tree with Matula-Goebel number 7 is the 3-edge rooted tree Y of height 2.

François Marques, Table of n, a(n) for n = 1..10000 (terms 1..5381 from Alois P. Heinz).
Emeric 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. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
Index entries for sequences related to Matula-Goebel numbers

A left inverse of A007097.
Cf. A003963, A061775, A091233.
Cf. A000081, A000720, A001222, A109129, A112798, A196050, A290822, A317713, A320325, A324927 (positions of 2), A324928 (positions of 3), A325032.
This statistic is counted by A034781, ordered A080936.
The ordered version is A358379.
For node-height instead of edge-height we have A358552.
Cf. A000108, A055277, A056239, A342507, A358576.

with(numtheory): a := proc(n) option remember; if n = 1 then 0 elif isprime(n) then 1+a(pi(n)) else max((map (p->a(p), factorset(n)))[]) end if end proc: seq(a(n), n = 1 .. 100); # Emeric Deutsch, Sep 16 2011

a [n_] := a[n] = If[n == 1, 0, If[PrimeQ[n], 1+a[PrimePi[n]], Max[Map[a, FactorInteger[n][[All, 1]]]]]]; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 06 2014, after Emeric Deutsch *)

a(n) = my(v=factor(n)[,1],d=0); while(#v,d++; v=fold(setunion, apply(p->factor(primepi(p))[,1]~, v))); d; \\ Kevin Ryde, Sep 21 2020

from functools import lru_cache
from sympy import isprime, primepi, primefactors
@lru_cache(maxsize=None)
def A109082(n):
    if n == 1 : return 0
    if isprime(n): return 1+A109082(primepi(n))
    return max(A109082(p) for p in primefactors(n)) # Chai Wah Wu, Mar 19 2022

a(1)=0; if n is the t-th prime, then a(n) = 1 + a(t); if n is composite, n=t*s, then a(n) = max(a(t),a(s)). The Maple program is based on this.
a(A007097(n)) = n.
a(n) = A358552(n) - 1. - Gus Wiseman, Nov 27 2022

Edited by Emeric Deutsch, Sep 16 2011

A324926 Numbers not divisible by any prime indices of their prime indices.

Original entry on oeis.org

1, 2, 4, 5, 8, 11, 16, 17, 22, 23, 25, 31, 32, 34, 41, 44, 47, 55, 59, 62, 64, 67, 73, 82, 83, 85, 88, 97, 103, 109, 115, 118, 121, 124, 125, 127, 128, 134, 137, 149, 157, 164, 166, 167, 176, 179, 187, 191, 194, 197, 205, 211, 218, 227, 233, 235, 236, 241, 242

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

A prime index of n is a number m such that prime(m) divides n. For example, the prime indices of 55 are {3,5} with prime indices {{2},{3}}. Since 55 is not divisible by 2 or 3, it belongs to the sequence.

			The sequence of multisets of multisets whose MM-numbers (see A302242) belong to the sequence begins:
   1: {}
   2: {{}}
   4: {{},{}}
   5: {{2}}
   8: {{},{},{}}
  11: {{3}}
  16: {{},{},{},{}}
  17: {{4}}
  22: {{},{3}}
  23: {{2,2}}
  25: {{2},{2}}
  31: {{5}}
  32: {{},{},{},{},{}}
  34: {{},{4}}
  41: {{6}}
  44: {{},{},{3}}
  47: {{2,3}}
  55: {{2},{3}}
  59: {{7}}
  62: {{},{5}}
  64: {{},{},{},{},{},{}}

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

Cf. A001222, A003963, A112798, A120383, A302242, A304360, A324846, A324847, A324848, A324849, A324850, A324927, A324928, A324930.

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

A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.

Original entry on oeis.org

1, 3, 5, 9, 13, 19, 25, 35, 45, 59, 73, 93, 113, 139, 165, 201, 237, 283, 329, 389, 449, 523, 597, 691, 785, 899, 1013, 1153, 1293, 1459, 1625, 1827, 2029, 2267, 2505, 2789, 3073, 3403, 3733, 4123, 4513, 4963, 5413, 5937, 6461, 7059, 7657, 8349

Offset: 1

Mitch Harris, Jan 05 2005

nonn

From Gus Wiseman, Mar 23 2019: (Start)
The offset could safely be changed to zero by setting the boundary condition to a(0) = 0.
Also the number of integer partitions of 2n into powers of 2 with at least one part > 1. The Heinz numbers of these partitions are given by A324927. For example, the a(1) = 1 through a(5) = 13 integer partitions are:
  (2)  (4)    (42)     (8)        (82)
       (22)   (222)    (44)       (442)
       (211)  (411)    (422)      (811)
              (2211)   (2222)     (4222)
              (21111)  (4211)     (4411)
                       (22211)    (22222)
                       (41111)    (42211)
                       (221111)   (222211)
                       (2111111)  (421111)
                                  (2221111)
                                  (4111111)
                                  (22111111)
                                  (211111111)
(End)

Cf. A000123, A033485.

Cf. A018819, A318400, A324927, A324928.

Table[Length[Select[IntegerPartitions[n],And[Max@@#>1,And@@IntegerQ/@Log[2,#]]&]],{n,0,30,2}] (* Gus Wiseman, Mar 23 2019 *)

from itertools import islice
from collections import deque
def A102378_gen(): # generator of terms
    aqueue, f, b, a = deque([2]), True, 1, 2
    yield from (1, 3)
    while True:
        a += b
        yield 2*a - 1
        aqueue.append(a)
        if f: b = aqueue.popleft()
        f = not f
A102378_list = list(islice(A102378_gen(),40)) # Chai Wah Wu, Jun 08 2022

a(n) - a(n-1) = A018819(n+1)
G.f. A(x) satisfies (1-x)*A(x) = 2(1 + x)*B(x^2), where B(x) is the gf of A033485
a(n) = A000123(n) - 1. - Gus Wiseman, Mar 23 2019
G.f. A(x) satisfies: A(x) = (x + (1 - x^2) * A(x^2)) / (1 - x)^2. - Ilya Gutkovskiy, Aug 11 2021

A324927 Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.

Original entry on oeis.org

3, 6, 7, 9, 12, 14, 18, 19, 21, 24, 27, 28, 36, 38, 42, 48, 49, 53, 54, 56, 57, 63, 72, 76, 81, 84, 96, 98, 106, 108, 112, 114, 126, 131, 133, 144, 147, 152, 159, 162, 168, 171, 189, 192, 196, 212, 216, 224, 228, 243, 252, 262, 266, 288, 294, 304, 311, 318

Offset: 1

Gus Wiseman, Mar 21 2019

nonn

Numbers n such that A109082(n) = 2.
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.
Also Heinz numbers of integer partitions into powers of 2 with at least one part > 1 (counted by A102378).

			The sequence of terms together with their prime indices begins:
   3: {2}
   6: {1,2}
   7: {4}
   9: {2,2}
  12: {1,1,2}
  14: {1,4}
  18: {1,2,2}
  19: {8}
  21: {2,4}
  24: {1,1,1,2}
  27: {2,2,2}
  28: {1,1,4}
  36: {1,1,2,2}
  38: {1,8}
  42: {1,2,4}
  48: {1,1,1,1,2}
  49: {4,4}
  53: {16}
  54: {1,2,2,2}
  56: {1,1,1,4}

Cf. A000081, A000720, A003963, A007097, A018819, A033844, A056239, A102378, A112798, A302242, A318400, A324928, A324929.

Select[Range[100],And[!IntegerQ[Log[2,#]],And@@Cases[FactorInteger[#],{p_,_}:>IntegerQ[Log[2,PrimePi[p]]]]]&]

A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 2, 0, 0, 1, 0, 1, 2, 0, 2, 1, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 0, 1, 1, 2, 0, 2, 1, 1, 2, 2, 0, 0, 2, 2, 1, 0, 0, 2, 0, 0, 1, 1, 1, 2, 1, 0, 0, 2, 1, 3, 2, 2, 1, 1, 0, 3, 1, 2, 0, 1, 1, 1, 1, 0, 2, 2, 0, 3, 2, 1

Offset: 1

Gus Wiseman, Mar 21 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 finite multiset of finite multisets of finite multisets of positive integers with MMM number n is obtained by factoring n into prime numbers, then factoring each of their prime indices into prime numbers, then factoring each of their prime indices into prime numbers, and finally taking their prime indices.

			The sequence of all finite multisets of finite multisets of finite multisets of positive integers begins (o is the empty multiset):
   1: o
   2: (o)
   3: ((o))
   4: (oo)
   5: (((1)))
   6: (o(o))
   7: ((oo))
   8: (ooo)
   9: ((o)(o))
  10: (o((1)))
  11: (((2)))
  12: (oo(o))
  13: ((o(1)))
  14: (o(oo))
  15: ((o)((1)))
  16: (oooo)
  17: (((11)))
  18: (o(o)(o))
  19: ((ooo))
  20: (oo((1)))

Cf. A000081, A000720, A001222, A050338, A056239, A112798, A301595, A302242, A318564, A318565, A318566, A324928.

fi[n_]:=If[n==1,{},FactorInteger[n]];
Table[Total[Cases[fi[n],{p_,k_}:>k*Total[Cases[fi[PrimePi[p]],{q_,j_}:>j*PrimeOmega[PrimePi[q]]]]]],{n,60}]

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A109082 Depth of rooted tree having Matula-Goebel number n.

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A324926 Numbers not divisible by any prime indices of their prime indices.

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A102378 a(n) = a(n-1) + a([n/2]) + 1, a(1) = 1.

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A324927 Matula-Goebel numbers of rooted trees of depth 2. Numbers that are not powers of 2 but whose prime indices are all powers of 2.

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A324930 Total weight of the multiset of multisets of multisets with MMM number n. Totally additive with a(prime(n)) = A302242(n).

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