A113620
Numbers whose 3 prime powers are a permutation of each other. Numbers with 3 distinct prime factors whose 3 exponents are a permutation of the 3 bases.
Original entry on oeis.org
21600, 36000, 48600, 121500, 169344, 225000, 337500, 395136, 857304, 3000564, 6690816, 19600000, 24532992, 37380096, 53782400, 59295096, 88942644, 122500000, 161980416, 171478296, 658834400, 774400000, 943130628, 1022754816, 2155524696, 2344190625, 4326400000
Offset: 1
21600 = 2^5 * 3^3 * 5^2
36000 = 2^5 * 3^2 * 5^3
48600 = 2^3 * 3^5 * 5^2
121500 = 2^2 * 3^5 * 5^3
169344 = 2^7 * 3^3 * 7^2
225000 = 2^3 * 3^2 * 5^5
337500 = 2^2 * 3^3 * 5^5
395136 = 2^7 * 3^2 * 7^3
857304 = 2^3 * 3^7 * 7^2
3000564 = 2^2 * 3^7 * 7^3
6690816 = 2^11 * 3^3 * 11^2
24532992 = 2^11 * 3^2 * 11^3
37380096 = 2^13 * 3^3 * 13^2
59295096 = 2^3 * 3^2 * 7^7
88942644 = 2^2 * 3^3 * 7^7
161980416 = 2^13 * 3^2 * 13^3
171478296 = 2^3 * 3^11 * 11^2
943130628 = 2^2 * 3^11 * 11^3
2155524696 = 2^3 * 3^13 * 13^2
2344190625 = 3^7 * 5^5 * 7^3
4594613625 = 3^7 * 5^3 * 7^5
6511640625 = 3^5 * 5^7 * 7^3
14010910524 = 2^2 * 3^13 * 13^3
25015118625 = 3^5 * 5^3 * 7^7
35452265625 = 3^3 * 5^7 * 7^5
69486440625 = 3^3 * 5^5 * 7^7
736820803125 = 3^11 * 5^5 * 11^3
3083660425988 = 2^2 * 3^3 * 11^11
3566212687125 = 3^11 * 5^3 * 11^5
15792626953125 = 3^5 * 5^11 * 11^3
20542440283992 = 2^3 * 3^2 * 11^11
212323095703125 = 3^3 * 5^11 * 11^5
8666341994809125 = 3^5 * 5^3 * 11^11
21807007674642216 = 2^3 * 3^2 * 13^13
24073172207803125 = 3^3 * 5^5 * 11^11
32710511511963324 = 2^2 * 3^3 * 13^13
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.
Original entry on oeis.org
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
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]).
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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 *)
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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)])
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# 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))
Showing 1-2 of 2 results.