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A285769 (Product of distinct prime factors)^(Product of prime exponents). 5
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 36, 13, 14, 15, 16, 17, 36, 19, 100, 21, 22, 23, 216, 25, 26, 27, 196, 29, 30, 31, 32, 33, 34, 35, 1296, 37, 38, 39, 1000, 41, 42, 43, 484, 225, 46, 47, 1296, 49, 100, 51, 676, 53, 216, 55, 2744, 57, 58, 59, 900, 61, 62, 441 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
COMMENTS
a(n) and A066638 differ at {36, 72, 100, 108, 144, ...}, i.e., for all n in A036785, since a(n) takes the product of the multiplicities of prime factors of n, while A066638 takes the maximum value of the multiplicities of prime factors of n. For these n, a(n) > A066638(n).
a(1) = 1 since 1 is the empty product; 1^1 = 1.
a(p) = p since omega(p) = A001221(p) = 1 thus p^1 = p.
a(p^m) = p^m since omega(p) = 1 thus p^m is maintained.
For squarefree n with omega(n) > 1, a(n) = n.
For n with omega(n) > 1 and at least one multiplicity m > 1, a(n) > n. In other words, let a(n) = k^m, where k is the product of the distinct prime factors of n and m is the product of the multiplicities of the distinct prime factors of n. a(n) > n for n in A126706 since there are 2 or more prime factors in k and m > 1.
Squarefree kernels of terms a(n) > n: {6, 6, 10, 6, 14, 6, 10, 22, 15, 6, 10, 26, 6, 14, 30, 21, ...}.
LINKS
FORMULA
a(n) = A007947(n)^A005361(n).
EXAMPLE
a(2) = 2 since (2)^(1) = 2^1 = 2.
a(6) = 6 since (2*3)^(1*1) = 6^1 = 6.
a(12) = 36 since (2*3)^(2*1) = 6^2 = 36.
a(30) = 30 since (2*3*5)^(1*1*1) = 30^1 = 30.
a(144) = 1679616 since (2*3)^(4*2) = 6^8 = 1679616.
MATHEMATICA
Array[Power @@ Map[Times @@ # &, Transpose@ FactorInteger@ #] &, 63] (* Michael De Vlieger, Apr 25 2017 *)
PROG
(Python)
from sympy import divisor_count, divisors
from sympy.ntheory.factor_ import core
def rad(n): return max(list(filter(lambda i: core(i) == i, divisors(n))))
def a(n): return rad(n)**divisor_count(n/rad(n)) # Indranil Ghosh, Apr 26 2017
CROSSREFS
Sequence in context: A161949 A004851 A066638 * A219209 A223474 A062279
KEYWORD
nonn,easy
AUTHOR
Michael De Vlieger, Apr 25 2017
STATUS
approved

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Last modified March 28 15:38 EDT 2024. Contains 371254 sequences. (Running on oeis4.)