A178212 -id:A178212 - cp's OEIS frontend

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

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

Alford Arnold, Jul 08 2003

nonn

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

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

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

Subsequence of A014613, A307341, A178212.

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 *)

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

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

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

More terms from Reinhard Zumkeller, Jul 25 2003

A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity).

Original entry on oeis.org

16, 24, 32, 36, 40, 48, 54, 56, 60, 64, 72, 80, 81, 84, 88, 90, 96, 100, 104, 108, 112, 120, 126, 128, 132, 135, 136, 140, 144, 150, 152, 156, 160, 162, 168, 176, 180, 184, 189, 192, 196, 198, 200, 204, 208, 210, 216, 220, 224, 225, 228, 232, 234, 240, 243

Offset: 1

N. J. A. Sloane

Complement of A037144: A001222(a(n)) > 3; A117358(a(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
Also numbers such that no permutation of all proper divisors exists with coprime adjacent elements: A178254(a(n)) = 0. - Reinhard Zumkeller, May 24 2010
Also, numbers that can be written as a product of at least two composites, i.e., admit a nontrivial factorization into composites. - Felix Fröhlich, Dec 22 2018

T. D. Noe, Table of n, a(n) for n=1..1000

Subsequence of A033942; A178212 is a subsequence.

Cf. A014613, A001055, A033273.

with(numtheory): A033987:=n->`if`(bigomega(n)>3, n, NULL): seq(A033987(n), n=1..300); # Wesley Ivan Hurt, May 26 2015

Select[Range[300],PrimeOmega[#]>3&] (* Harvey P. Dale, Mar 20 2016 *)

is(n)=bigomega(n)>3 \\ Charles R Greathouse IV, May 26 2015

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A033987(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b,isqrt(x//c)+1),a)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b,integer_nthroot(x//c,m)[0]+1),a) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+primepi(x)+sum(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,i)) for i in range(2,4)))
    kmin, kmax = 1,2
    while f(kmax) >= kmax:
        kmax <<= 1
    while True:
        kmid = kmax+kmin>>1
        if f(kmid) < kmid:
            kmax = kmid
        else:
            kmin = kmid
        if kmax-kmin <= 1:
            break
    return kmax # Chai Wah Wu, Aug 23 2024

Product p_i^e_i with Sum e_i >= 4.
A001055(a(n)) > A033273(a(n)). - Juri-Stepan Gerasimov, Nov 09 2009
a(n) ~ n. - Charles R Greathouse IV, Jul 11 2024

More terms from Patrick De Geest, Jun 15 1998

A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 420, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492, 495

Offset: 1

Michael De Vlieger, Oct 22 2024

Also, numbers k such that there exists a pair of necessarily composite divisors {d, k/d}, d < k/d, with quality Q, i.e., gcd(d, k/d) > 1 but there exists a prime p | d that does not divide k/d, and also a prime q | k/d that does not divide d.
A178212 is a proper subset.
This sequence is distinct from A123712 since 420 is here.
This sequence is distinct from A182855 since 360 is here.

			a(1) = 60 = 2^2 * 3 * 5, the smallest number such that bigomega(60) > omega(60) > 2. Bigomega(60) = 4, omega(60) = 3.
72 is not in the sequence because it is the product of 2 distinct prime factors.
a(2) = 84 = 2^2 * 3 * 7, since bigomega(84) = 4, omega(84) = 3.
a(3) = 90 = 2 * 3^2 * 5, since bigomega(90) = 4, omega(90) = 3.
a(4) = 120 = 2^3 * 3 * 5, since bigomega(120) = 5, omega(120) = 3.
210 is not in the sequence because it is squarefree.
a(35) = 360 = 2^3 * 3^2 * 5 since bigomega(360) = 6, omega(360) = 3.
a(43) = 420 = 2^2 * 3 * 5 * 7 since bigomega(420) = 5, omega(420) = 4, etc.
.
Table showing pairs of factors of a(n) for select n, such that the pair possesses quality Q (see comments).
    n    a(n)   pair of factors with quality Q.
  -------------------------------------------------------------------
    1     60    6 X 10;
    2     84    6 X 14;
    3     90    6 X 15;
    4    120    6 X 20,  10 X 12;
    5    126    6 X 21;
    6    132    6 X 22;
    7    140   10 X 14;
    8    150   10 X 15;
   17    240    6 X 40,  10 X 24, 12 X 20;
   51    480    6 X 80,  10 X 48, 12 X 40, 20 X 24;
  117    840    6 X 140, 10 X 84, 12 X 70, 14 X 60, 20 X 42, 28 X 30.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A001221, A001222, A013929, A024619, A126706, A178212.

Select[Range[500], PrimeOmega[#] > PrimeNu[#] > 2 &]

{a(n)} = { k : bigomega(k) > omega(k) > 2 }, where bigomega = A001222 and omega = A001221.

A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

Original entry on oeis.org

60, 84, 90, 120, 126, 132, 140, 150, 156, 168, 180, 198, 204, 220, 228, 234, 240, 252, 260, 264, 270, 276, 280, 294, 300, 306, 308, 312, 315, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 396, 408, 414, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Paul D. Hanna, Oct 09 2006

nonn

Triangle A123706 is the matrix inverse of triangle A010766, where A010766(n,k) = [n/k].
a(n) = A178212(n) for n <= 52, possibly more. [Reinhard Zumkeller, May 24 2010]
a(n) = A178212(n) for n <= 2000. - Bill McEachen, Jul 14 2024

G. C. Greubel, Table of n, a(n) for n = 1..1200

Cf. A123706, A123709, A123710, A123711, A010766.

Moebius[i_, j_] := If[Divisible[i, j], MoebiusMu[i/j], 0]; A123709[n_] := Length[Select[Table[Moebius[n, j] - Moebius[n, j + 1], {j, 1, n}], # != 0 &]]; Select[Range[6500], A123709[#] == 16 &] (* G. C. Greubel, Apr 22 2017 *)

is(n)=my(M=matrix(n, n, r, c,r\c)^-1); sum(k=1, n, M[n, k]!=0)==16 \\ Charles R Greathouse IV, Feb 09 2012

A200511 Numbers n with omega(n)=2 and bigomega(n)>2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity.

Original entry on oeis.org

12, 18, 20, 24, 28, 36, 40, 44, 45, 48, 50, 52, 54, 56, 63, 68, 72, 75, 76, 80, 88, 92, 96, 98, 99, 100, 104, 108, 112, 116, 117, 124, 135, 136, 144, 147, 148, 152, 153, 160, 162, 164, 171, 172, 175, 176, 184, 188, 189, 192, 196, 200, 207, 208, 212, 216, 224, 225, 232, 236

Offset: 1

M. F. Hasler, Feb 09 2012

nonn

Equivalently, numbers of the form n=p^k*q^m where k,m>0, k+m>2 and p,q prime.

It appears that this is equal to A123711.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A200521, A178212.

Select[Range[240], PrimeNu[#] == 2 && PrimeOmega[#] > 2 &] (* Jean-François Alcover, Jun 29 2013 *)

for(n=1,999,bigomega(n)>2 & omega(n)==2 & print1(n","))

A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.

Original entry on oeis.org

420, 630, 660, 780, 840, 924, 990, 1020, 1050, 1092, 1140, 1170, 1260, 1320, 1380, 1386, 1428, 1470, 1530, 1540, 1560, 1596, 1638, 1650, 1680, 1710, 1716, 1740, 1820, 1848, 1860, 1890, 1932, 1950, 1980, 2040, 2070, 2100, 2142, 2184, 2220, 2244

Offset: 1

M. F. Hasler, Feb 09 2012

nonn

I expect that A123709(a(k))=32.

M. F. Hasler, Table of n, a(n) for n = 1..18347

Cf. A200511, A178212.

Select[Range[2500], PrimeNu[#] == 4 && PrimeOmega[#] > 4 &](* Jean-François Alcover, Jun 30 2013 *)

is_A200521(n,c=4)={ omega(n)==c & bigomega(n)>c }

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

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A033987 Numbers that are divisible by at least 4 primes (counted with multiplicity).

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A375055 Nonsquarefree numbers k divisible by at least 3 distinct primes.

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A123712 Indices n such that 16 = A123709(n) = number of nonzero terms in row n of triangle A123706.

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A200511 Numbers n with omega(n)=2 and bigomega(n)>2, where omega=A001221=number of distinct prime factors, bigomega=A001222=prime factors counted with multiplicity.

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A200521 Numbers n such that omega(n)=4 but bigomega(n)>4, i.e., having exactly 4 distinct prime factors, but at least one of these with multiplicity > 1.

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