A231293
Numbers m such that, in the prime factorization of m, the product of the prime factors equals the sum of prime factors and exponents.
Original entry on oeis.org
20, 50, 112, 392, 1372, 2816, 3645, 4802, 6075, 10125, 13312, 15488, 16875, 28125, 46875, 85184, 86528, 278528, 413343, 468512, 562432, 964467, 1245184, 2250423, 2367488, 2576816, 3655808, 3932160, 5250987, 5898240, 8847360, 9830400, 11829248, 12252303
Offset: 1
50 = 2 * 5^2; the product of the prime factors is 2 * 5 = 10, the sum of the prime factors and exponents is 2 + 1 + 5 + 2 = 10, hence 50 is in the sequence.
112 = 2^4 * 7; the product of the prime factors is 2 * 7 = 14, the sum of the prime factors and exponents is 2 + 4 + 7 + 1 = 14, hence 112 is in the sequence.
14172488 = 2^3 * 11^6, product of prime factors is 2*11 = 22, sum of prime factors and exponents is 2 + 3 + 11 + 6 = 22, hence 14172488 is in the sequence.
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t = {}; n = 1; While[Length[t] < 30, n++; f = FactorInteger[n]; sm = Total[Flatten[f]]; pr = Times @@ Transpose[f][[1]]; If[sm == pr, AppendTo[t, n]]]; t
ppfQ[n_]:=Module[{f=FactorInteger[n]},Times@@[f][[All,1]] == Total[ Flatten[f]]]; Select[Range[13*10^6],ppfQ] (* Harvey P. Dale, Aug 17 2016 *)
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))
A356433
Numbers k such that, in the prime factorization of k, the least common multiple of the exponents equals the least common multiple of the prime factors.
Original entry on oeis.org
1, 4, 27, 72, 108, 192, 576, 800, 1458, 1728, 2916, 3125, 5120, 5832, 6272, 12500, 21600, 25600, 30375, 36000, 46656, 48600, 77760, 84375, 114688, 116640, 121500, 138240, 169344, 225000, 247808, 337500, 384000, 388800, 395136, 583200, 600000, 653184, 691200, 750141, 802816, 823543, 857304, 979776
Offset: 1
576 = 2^6 * 3^2, lcm(2,3) = 6 = lcm(6,2), hence 576 is a term.
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Select[Range[10^6], Equal @@ LCM @@ FactorInteger[#] &] (* Amiram Eldar, Aug 07 2022 *)
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isok(k) = my(f=factor(k)); lcm(f[,1]) == lcm(f[,2]); \\ Michel Marcus, Aug 07 2022
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from math import lcm
from sympy import factorint
def ok(n): f = factorint(n); return lcm(*f.keys()) == lcm(*f.values())
print([k for k in range(10**6) if ok(k)]) # Michael S. Branicky, Aug 07 2022
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