cp's OEIS Frontend

A179643 Products of exactly 2 distinct squares of primes and a different prime (p^2 * q^2 * r).

%I A179643 #32 Apr 22 2025 04:11:33
%S A179643 180,252,300,396,450,468,588,612,684,700,828,882,980,1044,1100,1116,
%T A179643 1300,1332,1452,1476,1548,1575,1692,1700,1900,1908,2028,2124,2156,
%U A179643 2178,2196,2205,2300,2412,2420,2450,2475,2548,2556,2628,2844,2900,2925,2988
%N A179643 Products of exactly 2 distinct squares of primes and a different prime (p^2 * q^2 * r).
%C A179643 A050326(a(n)) = 5, subsequence of A225228. - _Reinhard Zumkeller_, May 03 2013
%H A179643 T. D. Noe, <a href="/A179643/b179643.txt">Table of n, a(n) for n = 1..1000</a>
%H A179643 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures">List of Prime Signatures</a>
%H A179643 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%e A179643 180 = 2^2 * 3^2 * 5, 252 = 2^2 * 3^2 * 7, 300 = 2^2 * 3 * 5^2, ...
%t A179643 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,2,2}; Select[Range[3000], f]
%o A179643 (PARI) list(lim)=my(v=List(),t);forprime(p=2,sqrt(lim\12),forprime(q=p+1,sqrt(lim\p^2\2),t=(p*q)^2;forprime(r=2,lim\t,if(p==r||q==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 19 2011
%o A179643 (Python)
%o A179643 from math import isqrt
%o A179643 from sympy import primepi, primerange, integer_nthroot
%o A179643 def A179643(n):
%o A179643     def bisection(f,kmin=0,kmax=1):
%o A179643         while f(kmax) > kmax: kmax <<= 1
%o A179643         kmin = kmax >> 1
%o A179643         while kmax-kmin > 1:
%o A179643             kmid = kmax+kmin>>1
%o A179643             if f(kmid) <= kmid:
%o A179643                 kmax = kmid
%o A179643             else:
%o A179643                 kmin = kmid
%o A179643         return kmax
%o A179643     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=isqrt(x//r))))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(x+1))+sum(primepi(isqrt(x//p**3)) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,5)[0])
%o A179643     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A179643 Cf. A006881, A007304, A065036, A085986, A085987, A178739, A179642.
%Y A179643 Cf. A179695.
%K A179643 nonn
%O A179643 1,1
%A A179643 _Vladimir Joseph Stephan Orlovsky_, Jul 21 2010