A285769 (Product of distinct prime factors)^(Product of prime exponents).

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

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

Michael De Vlieger, Table of n, a(n) for n = 1..10000

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

Cf. A005361, A007947, A066638, A088865, A126706.

Sequence in context: A161949 A004851 A066638 * A219209 A223474 A062279

Adjacent sequences: A285766 A285767 A285768 * A285770 A285771 A285772

Keyword

nonn,easy

Author

Michael De Vlieger, Apr 25 2017

Status

approved