cp's OEIS Frontend

A189975 Numbers with prime factorization pqr^3 for distinct p, q, r.

%I A189975 #35 Apr 22 2025 04:33:07
%S A189975 120,168,264,270,280,312,378,408,440,456,520,552,594,616,680,696,702,
%T A189975 728,744,750,760,888,918,920,945,952,984,1026,1032,1064,1128,1144,
%U A189975 1160,1240,1242,1272,1288,1416,1464,1480,1485,1496,1566,1608,1624,1640,1672
%N A189975 Numbers with prime factorization pqr^3 for distinct p, q, r.
%H A189975 T. D. Noe, <a href="/A189975/b189975.txt">Table of n, a(n) for n = 1..1000</a>
%H A189975 Will Nicholes, <a href="https://willnicholes.com/2010/06/06/list-of-prime-signatures/">List of prime signatures</a>
%H A189975 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%t A189975 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,3}; Select[Range[2000],f]
%o A189975 (PARI) list(lim)=my(v=List(),t);forprime(p=2,(lim\6)^(1/3),forprime(q=2,sqrt(lim\p^3),if(p==q,next);t=p^3*q;forprime(r=q+1,lim\t,if(p==r,next);listput(v,t*r))));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 19 2011
%o A189975 (Python)
%o A189975 from math import isqrt
%o A189975 from sympy import primepi, primerange, integer_nthroot
%o A189975 def A189975(n):
%o A189975     def bisection(f,kmin=0,kmax=1):
%o A189975         while f(kmax) > kmax: kmax <<= 1
%o A189975         kmin = kmax >> 1
%o A189975         while kmax-kmin > 1:
%o A189975             kmid = kmax+kmin>>1
%o A189975             if f(kmid) <= kmid:
%o A189975                 kmax = kmid
%o A189975             else:
%o A189975                 kmin = kmid
%o A189975         return kmax
%o A189975     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**3)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(integer_nthroot(x,3)[0]+1))+sum(primepi(x//p**4) for p in primerange(integer_nthroot(x,4)[0]+1))-primepi(integer_nthroot(x,5)[0])
%o A189975     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A189975 Cf. A030634, A050997, A178739.
%K A189975 nonn
%O A189975 1,1
%A A189975 _Vladimir Joseph Stephan Orlovsky_, May 03 2011