%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