A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree.

6, 10, 14, 15, 21, 22, 24, 26, 30, 33, 34, 35, 38, 39, 40, 42, 46, 51, 54, 55, 56, 57, 58, 60, 62, 65, 66, 69, 70, 74, 77, 78, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119, 120, 122, 123, 126, 129, 130, 132, 133, 134, 135, 136, 138, 140, 141, 142, 143, 145, 146, 150

Offset: 1

Author

Chris Boyd, Sep 15 2013

Keywords

• nonn

Comments

Subsequence of A048943. According to Gerard P. Michon, one of the criteria for N to belong to A048943 is that it has at least two prime factors with odd multiplicities. By definition, the composite factor c in any term of A229153 conforms to this criterion.

From a(1) to a(63), identical to the given terms of A119847, except for the single term a(55) = 120.

Links

• Chris Boyd, Table of n, a(n) for n = 1..10000

Crossrefs

Complement of A265640.

Cf. A048943, A229125.

Programs

PARI

iscomposite(n)={if(!isprime(n)&&n!=1,return(1));}
test(n)={if(iscomposite(core(n)),return(1));}
for(n=1,200,if(test(n)==1,print1(n",")))

PARI

lista(nn) = {for(n=1,nn, if(!ispseudoprime(core(n)) && !issquare(n), print1(n, ", ")));} \\ Altug Alkan, Feb 04 2016

PARI

list(lim)=my(v=List()); forsquarefree(c=6,lim\=1, if(#c[2]~ > 1, for(m=1,sqrtint(lim\c[1]), listput(v, c[1]*m^2)))); Set(v) \\ Charles R Greathouse IV, Jan 09 2022

Python

from math import isqrt
from sympy import primepi, mobius
def A229153(n):
    def f(x):
        c = n+x+(a:=isqrt(x))
        for y in range(1,a+1):
            m = x//y**2
            c += primepi(m)-sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 30 2025