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.

1, 4, 27, 72, 108, 800, 3125, 6272, 12500, 30375, 36000, 48600, 84375, 247808, 337500, 395136, 750141, 823543, 857304, 1384448, 3294172, 22235661, 24532992, 37879808, 53782400, 88942644, 122500000, 161980416, 171478296, 189267968, 235782657, 600112800, 1313046875, 2155524696

Offset

1,2

Links

Table of n, a(n) for n=1..34.

Example

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]).
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]).
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]).
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]).

Mathematica

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 *)

Prog

(Sage)
def in_seq( n ):
    if n == 1: return True
    pf = list( factor( n ) )
    primes    = [ t[0] for t in pf ]
    exponents = [ t[1] for t in pf ]
    if primes[0] in exponents:
        i = exponents.index(primes[0])
        exp_rotated = exponents[i : ] + exponents[0 : i]
        return primes == exp_rotated
    else:
        return False
print([n for n in range(1, 10000000) if in_seq(n)])
(Sage)
# Much faster program that generates the solutions rather than searching for them.
from sage.misc.misc import powerset
primes = primes_first_n(9)
max_prime = primes[-1]
solutions = set([1])
max_solution = min(2^max_prime * max_prime^2, max_prime^max_prime)
for subset in powerset(primes):
    subset_list = list(subset)
    for i in range(0, len(subset_list)):
        exponents = subset_list[i : ] + subset_list[0 : i]
        product = 1
        for j in range(0, len(subset_list)):
            product = product * subset_list[j]^exponents[j]
        if product <= max_solution:
            solutions.add(product)
print(sorted(solutions))

Crossrefs

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

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

Sequence in context: A054412 A122405 A122406 * A071837 A334633 A266011

Adjacent sequences: A276369 A276370 A276371 * A276373 A276374 A276375

Keyword

nonn

Author

Robert C. Lyons, Aug 31 2016

Status

approved