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

1,2

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

Chai Wah Wu, Algorithms for Complementary Sequences, Integers (2025) Vol. 25, Art. No. A95. See p. 21.

Example

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).

Mathematica

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

Prog

(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

Crossrefs

Complement of A126706.

Union of A005117 and A246547.

Union of A000469 and A246655.

Union of A000961 and A120944.

Cf. A025475.

Sequence in context: A359889 A236510 A317710 * A325328 A316521 A085156

Adjacent sequences: A303551 A303552 A303553 * A303555 A303556 A303557

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 26 2018

Status

approved