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A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).

%I A131605 #30 Aug 15 2025 12:49:02
%S A131605 1,36,100,144,196,216,225,324,400,441,484,576,676,784,900,1000,1089,
%T A131605 1156,1225,1296,1444,1521,1600,1728,1764,1936,2025,2116,2304,2500,
%U A131605 2601,2704,2744,2916,3025,3136,3249,3364,3375,3600,3844,3969,4225,4356,4624
%N A131605 Perfect powers of nonprimes (m^k where m is a nonprime positive integer and k >= 2).
%C A131605 Although 1 is a square, is a cube, and so on..., 1 is sometimes excluded from perfect powers since it is not a well-defined power of 1 (1 = 1^k for any k in [2, 3, 4, 5, ...])
%C A131605 From _Michael De Vlieger_, Aug 11 2025: (Start)
%C A131605 This sequence is A001597 \ A246547, i.e., perfect powers without proper prime powers.
%C A131605 Union of {1} with the intersection of A001597 and A126706, where A126706 is the sequence of numbers that are neither prime powers nor squarefree.
%C A131605 Union of {1} and A286708 \ A052486, i.e., powerful numbers that are not prime powers, without Achilles numbers, but including the empty product. (End)
%H A131605 Michael De Vlieger, <a href="/A131605/b131605.txt">Table of n, a(n) for n = 1..10000</a> (Terms a(n), n = 1..1323 from Klaus Brockhaus, and n = 1324..8649 from Daniel Forgues.)
%F A131605 Sum_{n>=1} 1/a(n) = 1 + A072102 - A136141 = 1.10130769935514973882... . - _Amiram Eldar_, Aug 15 2025
%t A131605 With[{nn = 2^20}, {1}~Join~Select[Union@ Flatten@ Table[a^2*b^3, {b, Surd[nn, 3]}, {a, Sqrt[nn/b^3]}], And[Length[#2] > 1, GCD @@ #2 > 1] & @@ {#, FactorInteger[#][[;; , -1]]} &] ] (* _Michael De Vlieger_, Aug 11 2025 *)
%o A131605 (PARI) isok(n) = if (n == 1, return (1), return (ispower(n, ,&np) && (! isprime(np)))); \\ _Michel Marcus_, Jun 12 2013
%o A131605 (Python)
%o A131605 from sympy import mobius, integer_nthroot, primepi
%o A131605 def A131605(n):
%o A131605     def f(x): return int(n-2+x+sum(mobius(k)*((a:=integer_nthroot(x,k)[0])-1)+primepi(a) for k in range(2,x.bit_length())))
%o A131605     kmin, kmax = 1,2
%o A131605     while f(kmax) >= kmax:
%o A131605         kmax <<= 1
%o A131605     while True:
%o A131605         kmid = kmax+kmin>>1
%o A131605         if f(kmid) < kmid:
%o A131605             kmax = kmid
%o A131605         else:
%o A131605             kmin = kmid
%o A131605         if kmax-kmin <= 1:
%o A131605             break
%o A131605     return kmax # _Chai Wah Wu_, Aug 14 2024
%Y A131605 Cf. A000961, A001597, A024619, A025475, A052486, A072102, A126706, A136141, A246547, A286708.
%K A131605 nonn
%O A131605 1,2
%A A131605 _Daniel Forgues_, May 27 2008