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

60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726

Offset: 1

Author

Alford Arnold, Jul 08 2003

Keywords

• nonn

Comments

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).

A050326(a(n)) = 4. - Reinhard Zumkeller, May 03 2013

Examples

			a(1) = 60 since 60 = 2*2*3*5 and has three distinct prime factors.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index to sequences related to prime signature

Crossrefs

Cf. A001248, A006881, A030078, A054753, A007304, A050997, A046387, A036035, A086974.

Subsequence of A014613, A307341, A178212.

Programs

Mathematica

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2}; Select[Range[2000], f] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
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 *)

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

PARI

is(n)=vecsort(factor(n)[,2]~)==[1,1,2] \\ Charles R Greathouse IV, Oct 19 2015

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A085987(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    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])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

Extensions

More terms from Reinhard Zumkeller, Jul 25 2003