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

1,1

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

Prog

(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

Crossrefs

Complement of A265640.

Cf. A048943, A229125.

Sequence in context: A339438 A268376 A067582 * A119847 A279458 A119899

Adjacent sequences: A229150 A229151 A229152 * A229154 A229155 A229156

Keyword

nonn

Author

Chris Boyd, Sep 15 2013

Status

approved