A342679 Number of steps for n to reach 1 or n by repeated application of A037916, or -1 if they are never reached.

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 1, 3, 3, 2, 1, 2, 2, 4, 2, 3, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 3, 3, 2, 3, 1, 3, 2, 3, 1, 3, 1, 2, 4, 3, 2, 3, 1, 2, 3, 2, 1, 2, 2, 2, 2, 2, 1

Offset

2,3

Comments

Let n = (p1^a1)*(p2^a2)*...*(pk^aj) be the prime-factorization of n >= 2, where primes are in ascending order and ai >= 1 for all i >= 1 and <= k, then form n' = a1 a2 a3 ... ak = A037916(n), the concatenation of the exponents. Repeat this process if n' != 1 and != n, otherwise stop.

When n is prime, a(n) = 1; when n is semiprime, a(n) = 2.

Does every n reach 1 by this process, or does there exist some n whose trajectory enters a cycle, i.e., we reach n again instead of 1?

No cycles for n <= 10^9. - Michael S. Branicky, Mar 21 2021

Links

Table of n, a(n) for n=2..89.

Example

3 = 3^1 -> 1, so a(3) = 1;
6 = 2^1 * 3^1 -> 11 = 11^1 -> 1, so a(6) = 2;
16 = 2^4 -> 4 = 2^2 -> 2 = 2^1 -> 1, so a(16) = 3;
50 = 2^1 * 5^2 -> 12 = 2^2 * 3^1 -> 21 = 3^1 * 7^1 -> 11 -> 1, so a(50) = 4.

Mathematica

Table[Length@Rest@Most@FixedPointList[FromDigits[Last/@FactorInteger@#]&, k], {k, 2, 100}] (* Giorgos Kalogeropoulos, Apr 01 2021 *)

Prog

(Python)
import sympy
N=int(input())
A342679_n=[]
for n in range(2, N+1):
    n_0=n
    steps=0
    while not sympy.isprime(n) :
        exponents=list(sympy.factorint(n).values())
        m=""
        for i in exponents:
            m=m+str(i)
        n=int(m)
        if n==n_0:
            break
        steps+=1
    A342679_n.append(steps+1)
print(A342679_n)
(Python)
def a(n):
  c, iter = 1, A037916(n)
  while iter != 1 and iter != n: c, iter = c+1, A037916(iter)
  return c
print([a(n) for n in range(2, 89)]) # Michael S. Branicky, Mar 20 2021
(PARI) f(n) = my(f=factor(n)[, 2], s=""); for(i=1, #f~, s=concat(s, Str(f[i]))); eval(s); \\ A037916
a(n) = my(k=n, nb=0); while (k != 1, k = f(k); nb++); nb; \\ Michel Marcus, Mar 18 2021

Crossrefs

Cf. A000040, A037916.

Sequence in context: A327400 A323303 A068108 * A337135 A113309 A348955

Adjacent sequences: A342676 A342677 A342678 * A342680 A342681 A342682

Keyword

nonn,base

Author

Devansh Singh, Mar 18 2021

Status

approved