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

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, 32, 33, 34, 35, 37, 38, 39, 41, 42, 43, 46, 47, 49, 51, 53, 55, 57, 58, 59, 61, 62, 64, 65, 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

Offset: 1

Author

Ilya Gutkovskiy, Apr 26 2018

Keywords

• nonn

Examples

			42 is in the sequence because 42 = 2*3*7 (3 distinct prime factors).
81 is in the sequence because 81 = 3^4 (4 prime factors, 1 distinct).
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Prime Power
• Eric Weisstein's World of Mathematics, Squarefree

Crossrefs

Complement of A126706.

Union of A005117 and A246547.

Union of A000469 and A246655.

Union of A000961 and A120944.

Cf. A025475.

Programs

Mathematica

Select[Range[110], PrimePowerQ[#] || SquareFreeQ[#] &]
Select[Range[110], PrimeNu[#] == 1 || PrimeNu[#] == PrimeOmega[#] &]

Python

from math import isqrt
from sympy import primepi, integer_nthroot, mobius
def A303554(n):
    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)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 19 2024