A369015 -id:A369015 - cp's OEIS frontend

A369099 Index of first occurrence of n in A369015; smallest number whose prime tower factorization tree has Matula-Göbel number n.

1, 2, 4, 6, 16, 12, 64, 30, 36, 48, 65536, 60, 4096, 192, 144, 210, 18446744073709551616, 180, 1073741824, 240, 576, 196608, 68719476736, 420, 1296, 12288, 900, 960, 281474976710656, 720

Offset: 1

Pontus von Brömssen, Jan 13 2024

nonn

After a(30), the sequence continues 2^65536, 2*3*5*7*11, 2^16*3^2, 2^64*3, 2^6*3^4, 2^2*3^2*5*7, 2^60, 2^30*3, 2^12*9, 2^4*3*5*7, 2^4096, ... .
a(n) can be determined recursively as follows. Let n = Product_{i>=1} p_i^e_i, where p_i is the i-th prime. Take f_1 >= f_2 >= ... >= f_k so that the number a(i) occurs e_i times for i >= 1. Then a(n) = Product_{i>=1} p_i^f_i.
All terms are in A025487 (products of primorials).

			Using the method described in the comments for n = 20 = p(1)^2*p(3)^1, the exponents f_i shall include the term a(1)=1 twice and the term a(3)=4 once, i.e., (f_1, f_2, f_3) = (4, 1, 1), so a(20) = p(1)^4*p(2)^1*p(3) = 240.

Index entries for sequences related to Matula-Göbel numbers.

Cf. A002110, A007097, A014221, A025487, A369015, A369137, A369138.

A permutation of A284456.

from sympy import factorint,nextprime,primepi
def A369099(n):
    f = {A369099(primepi(p)):e for p,e in factorint(n).items()}
    a = p = 1
    for k in sorted(f,reverse=True):
        for i in range(f[k]):
            p = nextprime(p)
            a *= p**k
    return a

a(prime(n)) = 2^a(n). As a consequence, a(A007097(n)) = A014221(n).
a(2^n) = A002110(n).
a(n) = A284456(A369137(n)) = A025487(A369138(A369137(n))).

A369137 Inverse permutation to A369136.

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 11, 7, 8, 9, 51, 10, 28, 14, 12, 15, 1602, 13, 194, 16, 18, 60, 307, 17, 23, 35, 20, 21, 681, 19

Offset: 1

Pontus von Brömssen, Jan 14 2024

A284456(a(n)) is the smallest number whose prime tower factorization tree has Matula-Göbel number n.

After a(31), the sequence continues 25, 71, 1726, 31, 22, 1304, 221, 44, 24, ?, 26, ?, 79, 27, 343, ?, 29, 47, 33, 1867, 50, ?, 32, 98, 34, 250, 739, ?, 30, ?, ?, 37, 42, 66, 91, ?, 1935, 381, 41, ... .

Index entries for sequences related to Matula-Göbel numbers.

Cf. A007097, A014221, A025488, A098719, A284456, A369015, A369099, A369136, A369138.

A369015(A284456(a(n))) = A369136(a(n)) = n.
A369099(n) = A284456(a(n)).
A369138(a(A007097(n))) = A025488(A014221(n-1)).
A369138(a(2^n)) = A098719(n+1).

A369136 Matula-Göbel number of the prime tower factorization tree of A284456(n).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 9, 10, 12, 7, 15, 18, 14, 16, 20, 24, 21, 30, 27, 28, 36, 25, 40, 32, 42, 45, 13, 48, 60, 35, 54, 50, 56, 26, 72, 63, 80, 84, 90, 70, 64, 75, 39, 81, 100, 49, 120, 96, 52, 11, 108, 112, 126, 105, 135, 144, 140, 78, 22, 168, 150, 98, 160

Offset: 1

Pontus von Brömssen, Jan 14 2024

nonn

A permutation of the positive integers; the positive integers in the order they appear in A369015.

Pontus von Brömssen, Table of n, a(n) for n = 1..10000
Index entries for sequences related to Matula-Göbel numbers.

Inverse permutation to A369137.

Cf. A284456, A369015, A369099.

a(n) = A369015(A284456(n)).

A369099(a(n)) = A284456(n).

A368899 Least integer which begins n consecutive integers with the same prime tower factorization tree.

Original entry on oeis.org

1, 2, 33, 19940, 136824, 630772, 30530822

Offset: 1

Roberto Conti, Jan 09 2024

The (unordered) prime tower tree for k having prime factorization k = Product p[i]^e[i] comprises a root vertex and beneath it child subtrees with tree numbers e[i].
a(n) is the smallest k such that A369015(k) = A369015(k+i) for 1 <= i < n.
a(n) <= A034173(n) since it demands equal exponents but here they only have to be isomorphic.

			For n=5, a(5) = 136824 = 2^3 * 3^1 * 5701^1 has tree structure
   136824
   / | \
  3  1  1
  |
  1
The structures of the 5 numbers 136824, ..., 136828 are isomorphic as rooted trees, for example
   136826
   / | \
  1  2  1
     |
     1

Roberto Conti, Pierluigi Contucci, and Vitalii Iudelevich, Bounds on tree distribution in number theory, arXiv:2401.03278 [math.NT], 2023. See Section 5 (p. 13).
Roberto Conti and Pierluigi Contucci, A ℕatural Avenue, arXiv:2204.08982 [math.NT], 2023.

Cf. A034173, A369015.

A369890 The number of divisors of the largest divisor of n whose exponents in its prime factorization are all powers of 2.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 5, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 10, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 10, 5, 4, 2, 12, 4, 4

Offset: 1

Amiram Eldar, Feb 15 2024

First differs from A369015 at n = 32.

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

Cf. A000005, A000079, A053644, A138302, A353897, A369015.

f[p_, e_] := 2^Floor[Log2[e]] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> 2^logint(x, 2) + 1, factor(n)[, 2]));

a(n) = A000005(A353897(n)).
Multiplicative with a(p^e) = A053644(e) + 1.
a(n) = 2 if and only if n is prime.
a(n) <= A000005(n), with equality if and only if n is in A138302.

cp's OEIS Frontend

A369099 Index of first occurrence of n in A369015; smallest number whose prime tower factorization tree has Matula-Göbel number n.

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A369137 Inverse permutation to A369136.

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A369136 Matula-Göbel number of the prime tower factorization tree of A284456(n).

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A368899 Least integer which begins n consecutive integers with the same prime tower factorization tree.

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A369890 The number of divisors of the largest divisor of n whose exponents in its prime factorization are all powers of 2.

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