A323785 Iteratively swap prime factors of n with their exponent (unless the exponent is 1), until a cycle is reached, and then record the largest term in the cycle.

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 16, 17, 16, 19, 20, 21, 22, 23, 27, 32, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 32, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 48, 128, 32, 51, 52, 53, 54, 55, 63, 57, 58, 59, 60, 61, 62, 63, 32, 65, 66, 67

Offset

1,2

Links

Gabriel Antonius, Table of n, a(n) for n = 1..10000

Example

For n=196, the iterative procedure evolves as follows: 196 = 2^2 * 7^2 --> 2^2 * 2^7 = 2^9 --> 9^2 = 3^4 --> 4^3 = 2^6 --> 6^2 = 3^2 * 2^2 --> 2^3 * 2^2 = 2^5 --> 5^2 --> 2^5 = 32.
Since the iterative procedure stabilizes in the cycle 2^5=32 <--> 5^2=25, of length 2, the largest member of which is 32, we get a(196)=32.

Mathematica

p[x_, y_] := If[y > 1, y^x, x^y]; f[n_] := Times @@ (p[First[#], Last[#]] & /@ FactorInteger[n]); a[n_] := Module[{k = n, s = {k}}, While[! MemberQ[s, (k = f[k])], AppendTo[s, k]]; ind = Position[s, _?(# == k &)][[1, 1]]; Max[s[[ind ;; -1]]]]; Array[a, 57] (* Amiram Eldar, Sep 02 2019 *)

Prog

(Python)
from sympy import factorint, prod
def a(n):
    np = swap_prime_factors_with_exponents(n)
    npp = swap_prime_factors_with_exponents(np)
    return max(n, np) if n == npp else a(npp)
def swap_prime_factors_with_exponents(n):
    return prod([(e**p if e > 1 else p) for p, e in factorint(n).items()])
print([a(n) for n in range(1, 70)]) # simplified by Jason Yuen, Jan 12 2026

Crossrefs

Cf. A008477.

Sequence in context: A187099 A284049 A063932 * A327626 A385136 A179464

Adjacent sequences: A323782 A323783 A323784 * A323786 A323787 A323788

Keyword

nonn

Author

Gabriel Antonius, Aug 31 2019

Status

approved