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Numbers n such that, in the prime factorization of n, the list of the exponents is a rotation of the list of the prime factors.
0

%I #29 Feb 27 2020 11:48:37

%S 1,4,27,72,108,800,3125,6272,12500,30375,36000,48600,84375,247808,

%T 337500,395136,750141,823543,857304,1384448,3294172,22235661,24532992,

%U 37879808,53782400,88942644,122500000,161980416,171478296,189267968,235782657,600112800,1313046875,2155524696

%N Numbers n such that, in the prime factorization of n, the list of the exponents is a rotation of the list of the prime factors.

%e 4 is in the sequence because the prime factorization of 4 is 2^2, and the list of exponents (i.e., [2]) is a rotation of the list of prime factors (i.e., [2]).

%e 36000 is in the sequence because the prime factorization of 36000 is 2^5 * 3^2 * 5^3, and the list of exponents (i.e., [5, 2, 3]) is a rotation of the list of prime factors (i.e., [2, 3, 5]).

%e 84 is not in the sequence because the prime factorization of 84 is 2^2 * 3^1 * 7^1, and the list of exponents (i.e., [2, 1, 1]) is not a rotation of the list of prime factors (i.e., [2, 3, 7]).

%e 21600 is not in the sequence because the prime factorization of 21600 is 2^5 * 3^3 * 5^2, and the list of exponents (i.e., [5, 3, 2]) is not a rotation of the list of prime factors (i.e., [2, 3, 5]).

%t Select[Range[10^6], Function[w, Total@ Boole@ Map[First@ w == # &, RotateLeft[Last@ w, #] & /@ Range[Length@ Last@ w]] > 0]@ Transpose@ FactorInteger@ # &] (* _Michael De Vlieger_, Sep 01 2016 *)

%o (Sage)

%o def in_seq( n ):

%o if n == 1: return True

%o pf = list( factor( n ) )

%o primes = [ t[0] for t in pf ]

%o exponents = [ t[1] for t in pf ]

%o if primes[0] in exponents:

%o i = exponents.index(primes[0])

%o exp_rotated = exponents[i : ] + exponents[0 : i]

%o return primes == exp_rotated

%o else:

%o return False

%o print([n for n in range(1, 10000000) if in_seq(n)])

%o (Sage)

%o # Much faster program that generates the solutions rather than searching for them.

%o from sage.misc.misc import powerset

%o primes = primes_first_n(9)

%o max_prime = primes[-1]

%o solutions = set([1])

%o max_solution = min(2^max_prime * max_prime^2, max_prime^max_prime)

%o for subset in powerset(primes):

%o subset_list = list(subset)

%o for i in range(0, len(subset_list)):

%o exponents = subset_list[i : ] + subset_list[0 : i]

%o product = 1

%o for j in range(0, len(subset_list)):

%o product = product * subset_list[j]^exponents[j]

%o if product <= max_solution:

%o solutions.add(product)

%o print(sorted(solutions))

%Y Subsequence of A122406 and of A056166. A048102 is a subsequence.

%Y Cf. A008475, A008478, A054411, A054412, A082949, A113855.

%K nonn

%O 1,2

%A _Robert C. Lyons_, Aug 31 2016