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a(n) = smallest multiple m of n such that there are at most m/n perfect powers <= m.
1

%I #43 Mar 29 2023 06:08:00

%S 1,2,3,20,45,60,77,96,117,180,209,276,312,378,420,464,510,594,646,700,

%T 756,814,874,936,1050,1118,1188,1260,1363,1440,1519,1632,1716,1836,

%U 1925,2016,2183,2394,2496,2600,2788,2898,3010,3124,3330

%N a(n) = smallest multiple m of n such that there are at most m/n perfect powers <= m.

%C A perfect power is a power k^e with e != 1 (see A001597).

%C No value of the sequence belongs to A001597 except the first value 1.

%H Martin Y. Champel, <a href="/A233410/b233410.txt">Table of n, a(n) for n = 1..9999</a>

%H New Scientist, <a href="https://www.newscientist.com/article/mg22029450-400-enigma-number-1777/">Enigma 1777. Powerful tombola</a>, by Susan Denham, November 2013.

%e n=1: a(1)=1 as the only perfect power <= 1 is 1.

%e n=2: a(2)=2 as the only perfect power <= 2 is 1, and 2=2*1.

%e n=3: a(3)=3 as the only perfect power <= 3 is 1, and 3=3*1.

%e n=4: a(4)=20 as (1,4,8,9,16) are 5 perfect powers smaller than 20 and 20 = 4*5.

%e n=5: a(5)=45 as (1,4,8,9,16,25,27,32,36) are 9 perfect powers smaller than 45 and 45 = 5*9.

%o (Python)

%o # requires python code of A001597

%o class A233410() :

%o def __init__(self) :

%o self.a001597 = A001597()

%o def perf_pows_up_to(self, n):

%o idx = 1

%o while True:

%o if self.a001597.at(idx) > n :

%o return idx-1

%o idx += 1

%o def at(self, n):

%o k = 1

%o while True :

%o m = k*n

%o if self.perf_pows_up_to(m) <= k:

%o return m

%o k += 1

%o a233410 = A233410()

%o for n in range(1,20):

%o print(a233410.at(n)) # _R. J. Mathar_, Mar 29 2023

%Y Cf. A001597 = list of perfect powers, A069623 = number of perfect powers <= n

%K nonn

%O 1,2

%A _Martin Y. Champel_, Dec 09 2013