A284600 a(n) = n/(largest prime power dividing n).

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 1, 2, 1, 4, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 1, 3, 2, 5, 4, 1, 2, 3, 5, 1, 6, 1, 4, 5, 2, 1, 3, 1, 2, 3, 4, 1, 2, 5, 7, 3, 2, 1, 12, 1, 2, 7, 1, 5, 6, 1, 4, 3, 10, 1, 8, 1, 2, 3, 4, 7, 6, 1, 5, 1, 2, 1, 12, 5, 2, 3, 8, 1, 10

Offset: 1

Author

Ilya Gutkovskiy, Mar 30 2017

Keywords

• nonn

Comments

a(n) = smallest positive number k such that n/k is a prime power.

Examples

			a(12) = 3 because 12 = 2^2*3 therefore 12/(largest prime power dividing 12) = 12/4 = 3.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Ilya Gutkovskiy, Extended graphical example

Crossrefs

Cf. A000961, A003557, A007913, A034699, A052126, A121289, A278568.

Has same beginning as A052128 and A114536 but is strictly different from those two sequences.

Programs

Maple

f:= n ->  n /max(map(t -> t[1]^t[2], ifactors(n)[2])):
f(1):= 1:
map(f, [$1..100]); # Robert Israel, Apr 09 2017

Mathematica

Join[{1}, Table[n/Last[Select[Divisors[n], PrimePowerQ[#1] &]], {n, 2, 90}]]

Python

from sympy import lcm
def a003418(n): return 1 if n<1 else lcm(range(1, n + 1))
def a(n):
    m=1
    while True:
        if a003418(m)%n==0: return m
        else: m+=1
print([n//a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

Formula

a(n) = n/A034699(n).

a(n) = 1 if n is a prime power (A000961).

a(n) = 2 if n is a twice odd prime power (A278568).