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A276372 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.

%I A276372 #29 Feb 27 2020 11:48:37
%S A276372 1,4,27,72,108,800,3125,6272,12500,30375,36000,48600,84375,247808,
%T A276372 337500,395136,750141,823543,857304,1384448,3294172,22235661,24532992,
%U A276372 37879808,53782400,88942644,122500000,161980416,171478296,189267968,235782657,600112800,1313046875,2155524696
%N A276372 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 A276372 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 A276372 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 A276372 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 A276372 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 A276372 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 A276372 (Sage)
%o A276372 def in_seq( n ):
%o A276372     if n == 1: return True
%o A276372     pf = list( factor( n ) )
%o A276372     primes    = [ t[0] for t in pf ]
%o A276372     exponents = [ t[1] for t in pf ]
%o A276372     if primes[0] in exponents:
%o A276372         i = exponents.index(primes[0])
%o A276372         exp_rotated = exponents[i : ] + exponents[0 : i]
%o A276372         return primes == exp_rotated
%o A276372     else:
%o A276372         return False
%o A276372 print([n for n in range(1, 10000000) if in_seq(n)])
%o A276372 (Sage)
%o A276372 # Much faster program that generates the solutions rather than searching for them.
%o A276372 from sage.misc.misc import powerset
%o A276372 primes = primes_first_n(9)
%o A276372 max_prime = primes[-1]
%o A276372 solutions = set([1])
%o A276372 max_solution = min(2^max_prime * max_prime^2, max_prime^max_prime)
%o A276372 for subset in powerset(primes):
%o A276372     subset_list = list(subset)
%o A276372     for i in range(0, len(subset_list)):
%o A276372         exponents = subset_list[i : ] + subset_list[0 : i]
%o A276372         product = 1
%o A276372         for j in range(0, len(subset_list)):
%o A276372             product = product * subset_list[j]^exponents[j]
%o A276372         if product <= max_solution:
%o A276372             solutions.add(product)
%o A276372 print(sorted(solutions))
%Y A276372 Subsequence of A122406 and of A056166. A048102 is a subsequence.
%Y A276372 Cf. A008475, A008478, A054411, A054412, A082949, A113855.
%K A276372 nonn
%O A276372 1,2
%A A276372 _Robert C. Lyons_, Aug 31 2016