cp's OEIS Frontend

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

%I A229153 #20 Jan 30 2025 17:36:49
%S A229153 6,10,14,15,21,22,24,26,30,33,34,35,38,39,40,42,46,51,54,55,56,57,58,
%T A229153 60,62,65,66,69,70,74,77,78,82,84,85,86,87,88,90,91,93,94,95,96,102,
%U A229153 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
%N A229153 Numbers of the form c * m^2, where m > 0 and c is composite and squarefree.
%C A229153 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.
%C A229153 From a(1) to a(63), identical to the given terms of A119847, except for the single term a(55) = 120.
%H A229153 Chris Boyd, <a href="/A229153/b229153.txt">Table of n, a(n) for n = 1..10000</a>
%o A229153 (PARI) iscomposite(n)={if(!isprime(n)&&n!=1,return(1));}
%o A229153 test(n)={if(iscomposite(core(n)),return(1));}
%o A229153 for(n=1,200,if(test(n)==1,print1(n",")))
%o A229153 (PARI) lista(nn) = {for(n=1,nn, if(!ispseudoprime(core(n)) && !issquare(n), print1(n, ", ")));} \\ _Altug Alkan_, Feb 04 2016
%o A229153 (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
%o A229153 (Python)
%o A229153 from math import isqrt
%o A229153 from sympy import primepi, mobius
%o A229153 def A229153(n):
%o A229153     def f(x):
%o A229153         c = n+x+(a:=isqrt(x))
%o A229153         for y in range(1,a+1):
%o A229153             m = x//y**2
%o A229153             c += primepi(m)-sum(mobius(k)*(m//k**2) for k in range(1, isqrt(m)+1))
%o A229153         return c
%o A229153     m, k = n, f(n)
%o A229153     while m != k: m, k = k, f(k)
%o A229153     return m # _Chai Wah Wu_, Jan 30 2025
%Y A229153 Complement of A265640.
%Y A229153 Cf. A048943, A229125.
%K A229153 nonn
%O A229153 1,1
%A A229153 _Chris Boyd_, Sep 15 2013