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A085987 Product of exactly four primes, three of which are distinct (p^2*q*r).

%I A085987 #45 Apr 22 2025 03:51:03
%S A085987 60,84,90,126,132,140,150,156,198,204,220,228,234,260,276,294,306,308,
%T A085987 315,340,342,348,350,364,372,380,414,444,460,476,490,492,495,516,522,
%U A085987 525,532,550,558,564,572,580,585,620,636,644,650,666,693,708,726
%N A085987 Product of exactly four primes, three of which are distinct (p^2*q*r).
%C A085987 A014613 is completely determined by A030514, A065036, A085986, A085987 and A046386 since p(4) = 5. (cf. A000041). More generally, the first term of sequences which completely determine the k-almost primes can be found in A036035 (a resorted version of A025487).
%C A085987 A050326(a(n)) = 4. - _Reinhard Zumkeller_, May 03 2013
%H A085987 T. D. Noe, <a href="/A085987/b085987.txt">Table of n, a(n) for n = 1..1000</a>
%H A085987 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%e A085987 a(1) = 60 since 60 = 2*2*3*5 and has three distinct prime factors.
%t A085987 f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2}; Select[Range[2000], f] (* _Vladimir Joseph Stephan Orlovsky_, May 03 2011 *)
%t A085987 pefp[{a_,b_,c_}]:={a^2 b c,a b^2 c,a b c^2}; Module[{upto=800},Select[ Flatten[ pefp/@Subsets[Prime[Range[PrimePi[upto/6]]],{3}]]//Union,#<= upto&]] (* _Harvey P. Dale_, Oct 02 2018 *)
%o A085987 (PARI) list(lim)=my(v=List(),t,x,y,z);forprime(p=2,lim^(1/4),t=lim\p^2;forprime(q=p+1,sqrtint(t),forprime(r=q+1,t\q,x=p^2*q*r;y=p*q^2*r;listput(v,x);if(y<=lim,listput(v,y);z=p*q*r^2;if(z<=lim,listput(v,z))))));vecsort(Vec(v)) \\ _Charles R Greathouse IV_, Jul 15 2011
%o A085987 (PARI) is(n)=vecsort(factor(n)[,2]~)==[1,1,2] \\ _Charles R Greathouse IV_, Oct 19 2015
%o A085987 (Python)
%o A085987 from math import isqrt
%o A085987 from sympy import primepi, primerange, integer_nthroot
%o A085987 def A085987(n):
%o A085987     def bisection(f,kmin=0,kmax=1):
%o A085987         while f(kmax) > kmax: kmax <<= 1
%o A085987         kmin = kmax >> 1
%o A085987         while kmax-kmin > 1:
%o A085987             kmid = kmax+kmin>>1
%o A085987             if f(kmid) <= kmid:
%o A085987                 kmax = kmid
%o A085987             else:
%o A085987                 kmin = kmid
%o A085987         return kmax
%o A085987     def f(x): return n+x+sum((t:=primepi(s:=isqrt(y:=x//r**2)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,4)[0])
%o A085987     return bisection(f,n,n) # _Chai Wah Wu_, Mar 27 2025
%Y A085987 Cf. A001248, A006881, A030078, A054753, A007304, A050997, A046387, A036035, A086974.
%Y A085987 Subsequence of A014613, A307341, A178212.
%K A085987 nonn
%O A085987 1,1
%A A085987 _Alford Arnold_, Jul 08 2003
%E A085987 More terms from _Reinhard Zumkeller_, Jul 25 2003