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A052485 Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.

%I A052485 #35 Sep 10 2024 11:03:47
%S A052485 2,3,5,6,7,10,11,12,13,14,15,17,18,19,20,21,22,23,24,26,28,29,30,31,
%T A052485 33,34,35,37,38,39,40,41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,57,
%U A052485 58,59,60,61,62,63,65,66,67,68,69,70,71,73,74,75,76,77,78,79,80,82,83,84
%N A052485 Weak numbers (i.e., not powerful (1)): there is a prime p where p|n is true but p^2|n is not true.
%H A052485 Reinhard Zumkeller, <a href="/A052485/b052485.txt">Table of n, a(n) for n = 1..10000</a>
%F A052485 A112526(a(n)) = 0. - _Reinhard Zumkeller_, Sep 16 2011
%F A052485 a(n) ~ n. - _Charles R Greathouse IV_, Jul 19 2012
%F A052485 a(n) = n + O(sqrt(n)). - _Charles R Greathouse IV_, Jul 08 2022
%F A052485 Sum_{n>=1} 1/a(n)^s = zeta(s) - zeta(2*s)*zeta(3*s)/zeta(6*s), for s > 1. - _Amiram Eldar_, May 13 2023
%t A052485 Select[Range[1000], Min[FactorInteger[#][[All, 2]]] <= 1 &] (* _Geoffrey Critzer_, Feb 11 2015 *)
%o A052485 (Haskell)
%o A052485 a052485 n = a052485_list !! (n-1)
%o A052485 a052485_list = filter ((== 0) . a112526) [1..]
%o A052485 -- _Reinhard Zumkeller_, Sep 16 2011
%o A052485 (PARI) is(n)=n>1 && vecmin(factor(n)[,2])==1 \\ _Charles R Greathouse IV_, Mar 19 2014
%o A052485 (PARI) is(n)=!ispowerful(n) \\ _Charles R Greathouse IV_, Sep 18 2015
%o A052485 (Python)
%o A052485 from math import isqrt
%o A052485 from sympy import integer_nthroot, factorint
%o A052485 def A052485(n):
%o A052485     def f(x): return int(n+sum(isqrt(x//k**3) for k in range(1, integer_nthroot(x, 3)[0]+1) if all(d<=1 for d in factorint(k).values())))
%o A052485     m, k = n, f(n)
%o A052485     while m != k: m, k = k, f(k)
%o A052485     return m # _Chai Wah Wu_, Sep 10 2024
%Y A052485 Cf. A001694 (complement), A112526. Not the same as A007916.
%K A052485 nonn,easy
%O A052485 1,1
%A A052485 _Henry Bottomley_, Mar 16 2000