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A303554 Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.

%I A303554 #13 Feb 16 2025 08:33:54
%S A303554 1,2,3,4,5,6,7,8,9,10,11,13,14,15,16,17,19,21,22,23,25,26,27,29,30,31,
%T A303554 32,33,34,35,37,38,39,41,42,43,46,47,49,51,53,55,57,58,59,61,62,64,65,
%U A303554 66,67,69,70,71,73,74,77,78,79,81,82,83,85,86,87,89,91,93,94,95,97,101,102,103,105,106,107,109,110
%N A303554 Union of the prime powers (p^k, p prime, k >= 0) and numbers that are the product of 2 or more distinct primes.
%H A303554 Michael De Vlieger, <a href="/A303554/b303554.txt">Table of n, a(n) for n = 1..10000</a>
%H A303554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimePower.html">Prime Power</a>
%H A303554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Squarefree.html">Squarefree</a>
%e A303554 42 is in the sequence because 42 = 2*3*7 (3 distinct prime factors).
%e A303554 81 is in the sequence because 81 = 3^4 (4 prime factors, 1 distinct).
%t A303554 Select[Range[110], PrimePowerQ[#] || SquareFreeQ[#] &]
%t A303554 Select[Range[110], PrimeNu[#] == 1 || PrimeNu[#] == PrimeOmega[#] &]
%o A303554 (Python)
%o A303554 from math import isqrt
%o A303554 from sympy import primepi, integer_nthroot, mobius
%o A303554 def A303554(n):
%o A303554     def f(x): return int(n+x-sum(primepi(integer_nthroot(x,k)[0]) for k in range(2,x.bit_length()))-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1)))
%o A303554     m, k = n, f(n)
%o A303554     while m != k:
%o A303554         m, k = k, f(k)
%o A303554     return m # _Chai Wah Wu_, Aug 19 2024
%Y A303554 Complement of A126706.
%Y A303554 Union of A005117 and A246547.
%Y A303554 Union of A000469 and A246655.
%Y A303554 Union of A000961 and A120944.
%Y A303554 Cf. A025475.
%K A303554 nonn
%O A303554 1,2
%A A303554 _Ilya Gutkovskiy_, Apr 26 2018