A033987 -id:A033987 - cp's OEIS frontend

A014613 Numbers that are products of 4 primes.

16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90, 100, 104, 126, 132, 135, 136, 140, 150, 152, 156, 184, 189, 196, 198, 204, 210, 220, 225, 228, 232, 234, 248, 250, 260, 276, 294, 296, 297, 306, 308, 315, 328, 330, 340, 342, 344, 348, 350, 351, 364, 372, 375, 376

Offset: 1

Eric W. Weisstein

nonn

T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, Math. Intell., Vol. 30, No. 2, Spring 2008.
Eric Weisstein's World of Mathematics, Almost Prime.

Cf. A033987, A114106 (number of 4-almost primes <= 10^n).

Sequences listing r-almost primes, that is, the n such that A001222(n) = r: A000040 (r = 1), A001358 (r = 2), A014612 (r = 3), this sequence (r = 4), A014614 (r = 5), A046306 (r = 6), A046308 (r = 7), A046310 (r = 8), A046312 (r = 9), A046314 (r = 10), A069272 (r = 11), A069273 (r = 12), A069274 (r = 13), A069275 (r = 14), A069276 (r = 15), A069277 (r = 16), A069278 (r = 17), A069279 (r = 18), A069280 (r = 19), A069281 (r = 20). - Jason Kimberley, Oct 02 2011

Select[Range[200], Plus @@ Last /@ FactorInteger[ # ] == 4 &] (* Vladimir Joseph Stephan Orlovsky, Apr 23 2008 *)
Select[Range[400], PrimeOmega[#] == 4&] (* Jean-François Alcover, Jan 17 2014 *)

isA014613(n) = bigomega(n)==4 \\ Michael B. Porter, Dec 13 2009

from sympy import factorint
def ok(n): return sum(factorint(n).values()) == 4
print([k for k in range(377) if ok(k)]) # Michael S. Branicky, Nov 19 2021

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A014613(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(x,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(x//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(x//(k*m))+1),b)))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 17 2024

Product p_i^e_i with Sum e_i = 4.
a(n) ~ 6n log n / (log log n)^3. - Charles R Greathouse IV, May 04 2013
a(n) = A078840(4,n). - R. J. Mathar, Jan 30 2019

More terms from Patrick De Geest, Jun 15 1998

A033942 Positive integers with at least 3 prime factors (counted with multiplicity).

Original entry on oeis.org

8, 12, 16, 18, 20, 24, 27, 28, 30, 32, 36, 40, 42, 44, 45, 48, 50, 52, 54, 56, 60, 63, 64, 66, 68, 70, 72, 75, 76, 78, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100, 102, 104, 105, 108, 110, 112, 114, 116, 117, 120, 124, 125, 126, 128, 130, 132, 135, 136, 138, 140, 144

Offset: 1

Jeff Burch

nonn

A001055(a(n)) > 2; e.g., for a(3)=18 there are 4 factorizations: 1*18 = 2*9 = 2*3*3 = 3*6. - Reinhard Zumkeller, Dec 29 2001
A001222(a(n)) > 2; A054576(a(n)) > 1. - Reinhard Zumkeller, Mar 10 2006
Also numbers such that no permutation of all divisors exists with coprime adjacent elements: A109810(a(n))=0. - Reinhard Zumkeller, May 24 2010
A211110(a(n)) > 3. - Reinhard Zumkeller, Apr 02 2012
A060278(a(n)) > 0. - Reinhard Zumkeller, Apr 05 2013
Volumes of rectangular cuboids with each side > 1. - Peter Woodward, Jun 16 2015
If k is a term then so is k*m for m > 0. - David A. Corneth, Sep 30 2020
Numbers k with a pair of proper divisors of k, (d1,d2), such that d1 < d2 and gcd(d1,d2) > 1. - Wesley Ivan Hurt, Jan 01 2021

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

Cf. A014612.
A101040(a(n))=0.
A033987 is a subsequence; complement of A037143. - Reinhard Zumkeller, May 24 2010
Subsequence of A080257.
See also A002808 for 'Composite numbers' or 'Divisible by at least 2 primes'.

a033942 n = a033942_list !! (n-1)
a033942_list = filter ((> 2) . a001222) [1..]
-- Reinhard Zumkeller, Oct 27 2011

with(numtheory): A033942:=n->`if`(bigomega(n)>2, n, NULL): seq(A033942(n), n=1..200); # Wesley Ivan Hurt, Jun 23 2015

Select[ Range[150], Plus @@ Last /@ FactorInteger[ # ] > 2 &] (* Robert G. Wilson v, Oct 12 2005 *)
Select[Range[150],PrimeOmega[#]>2&] (* Harvey P. Dale, Jun 22 2011 *)

is(n)=bigomega(n)>2 \\ Charles R Greathouse IV, May 04 2013

from sympy import factorint
def ok(n): return sum(factorint(n).values()) > 2
print([k for k in range(145) if ok(k)]) # Michael S. Branicky, Sep 10 2022

from math import isqrt
from sympy import primepi, primerange
def A033942(n):
    def f(x): return int(n+primepi(x)+sum(primepi(x//k)-a for a,k in enumerate(primerange(isqrt(x)+1))))
    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

Numbers of the form Product p_i^e_i with Sum e_i >= 3.

a(n) ~ n. - Charles R Greathouse IV, May 04 2013

Corrected by Patrick De Geest, Jun 15 1998

A037144 Numbers with at most 3 prime factors (counted with multiplicity).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86

Offset: 1

N. J. A. Sloane

nonn

Complement of A033987: A001222(a(n))<=3; A117358(a(n))=1. - Reinhard Zumkeller, Mar 10 2006

Also numbers such that exist permutations of all proper divisors only with coprime adjacent elements: A178254(a(n))>0. - Reinhard Zumkeller, May 24 2010

Klaus Brockhaus, Table of n, a(n) for n = 1..10000

A037143 is a subsequence.

Cf. A033987, A001222, A117358, A128644.

[ n: n in [1..86] | n eq 1 or &+[ t[2]: t in Factorization(n) ] le 3 ]; /* Klaus Brockhaus, Mar 20 2007 */

Select[Range[100],PrimeOmega[#]<4&] (* Harvey P. Dale, Oct 15 2015 *)

is(n)=bigomega(n)<4 \\ Charles R Greathouse IV, Sep 14 2015

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A037144(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+x-2-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

a(n) ~ 2n log n/(log log n)^2. - Charles R Greathouse IV, Sep 14 2015

More terms from Reinhard Zumkeller, Mar 10 2006

More terms from Klaus Brockhaus, Mar 20 2007

A178254 Number of permutations of the proper divisors of n such that no adjacent elements have a common divisor greater than 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 6, 1, 2, 2, 6, 1, 4, 1, 6, 6, 0, 1, 4, 1, 4, 6, 6, 1, 0, 2, 6, 2, 4, 1, 36, 1, 0, 6, 6, 6, 0, 1, 6, 6, 0, 1, 36, 1, 4, 4, 6, 1, 0, 2, 4, 6, 4, 1, 0, 6, 0, 6, 6, 1, 0, 1, 6, 4, 0, 6, 36, 1, 4, 6, 36, 1, 0, 1, 6, 4, 4, 6, 36, 1, 0, 0, 6, 1, 0, 6, 6, 6, 0, 1, 0, 6, 4, 6, 6, 6, 0, 1, 4, 4, 0, 1, 36, 1

Offset: 1

Reinhard Zumkeller, May 24 2010

nonn

Depends only on prime signature;
range = {0, 1, 2, 4, 6, 36};
a(A033987(n)) = 0; a(A037144(n)) > 0;
a(A008578(n))=1; a(A168363(n))=2; a(A054753(n))=4; a(A006881(n))=6; a(A007304(n))=36.

			Proper divisors for n=21 are: 1, 3, and 7:
a(39) = #{[1,3,7], [1,7,3], [3,1,7], [3,7,1], [7,1,3], [7,3,1]} = 6;
proper divisors for n=12 are: 1, 2, 3, 4, and 6:
a(12) = #{[2,3,4,1,6], [4,3,2,1,6], [6,1,2,3,4], [6,1,4,3,2]} = 4;
proper divisors for n=42: 1, 2, 3, 6, 7, 14, and 21:
a(42) = #{[2,21,1,6,7,3,14], [2,21,1,14,3,7,6], [3,14,1,6,7,2,21], [3,14,1,21,2,7,6], [6,1,14,3,7,2,21], [6,1,21,2,7,3,14], ...} = 36, see the appended file for the list of all permutations.

R. Zumkeller, Table of n, a(n) for n = 1..10000
R. Zumkeller, Example: n=42

Cf. A109810.

A046304 Divisible by at least 5 primes (counted with multiplicity).

Original entry on oeis.org

32, 48, 64, 72, 80, 96, 108, 112, 120, 128, 144, 160, 162, 168, 176, 180, 192, 200, 208, 216, 224, 240, 243, 252, 256, 264, 270, 272, 280, 288, 300, 304, 312, 320, 324, 336, 352, 360, 368, 378, 384, 392, 396, 400, 405, 408, 416, 420, 432, 440, 448, 450, 456

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A033987.

Cf. A014614.

Select[Range[500],PrimeOmega[#]>4&] (* Harvey P. Dale, Apr 16 2013 *)

is(n)=bigomega(n)>4 \\ Charles R Greathouse IV, Sep 17 2015

from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A046304(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 almostprimepi(n, k):
        if k==0: return int(n>=1)
        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)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
    def f(x): return n+1+sum(almostprimepi(x,k) for k in range(1,5))
    return bisection(f,n,n) # Chai Wah Wu, Mar 29 2025

Product p_i^e_i with Sum e_i >= 5.

a(n) = n + O(n (log log n)^3/log n). - Charles R Greathouse IV, Apr 07 2017

A117358 a(n) = A032742(A032742(A032742(n))) = ((n/lpf(n))/lpf(n/lpf(n)))/lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 6, 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 1, 5, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 10, 3, 1, 1, 7, 1, 1, 1, 11, 1, 5, 1, 1, 1, 1, 1, 12, 1, 1, 1, 5, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Mar 10 2006

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537
Eric Weisstein's World of Mathematics, Least Prime Factor

Cf. A014673, A032742, A037144, A033987, A054576, A115561.

f[n_] := n/FactorInteger[n][[1, 1]]; (* f is A032742 *)
a[n_] := f@ f@ f@ n;
Array[a, 100] (* Jean-François Alcover, Dec 09 2021 *)
Table[Nest[#/FactorInteger[#][[1,1]]&,n,3],{n,110}] (* Harvey P. Dale, Oct 10 2024 *)

(define (A117358 n) (A032742 (A032742 (A032742 n)))) ;; Antti Karttunen, Dec 07 2017

a(n) = A032742(A032742(A032742(n))) = A032742(A054576(n)) = A054576(n)/A115561(n).

a(A037144(n)) = 1, a(A033987(n)) > 1.

A178212 Nonsquarefree numbers divisible by exactly three 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, 440, 444, 450, 456, 460, 468, 476, 480, 490, 492

Offset: 1

Reinhard Zumkeller, May 24 2010

nonn

			60 is in the sequence because it is not squarefree and it is divisible by three distinct primes: 2, 3, 5.
72 is not in the sequence, because although it is not squarefree, it is divisible by only two distinct primes: 2 and 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

A subsequence of A033987.
A085987 is a subsequence.
Cf. A007304, A200511, A200521.
Cf. A000005, A001221, A001222, A013929, A085987, A123712.
Cf. A027748, A124010.
Subsequence of A375055, which differs starting at a(43) = 440 > A375055(43) = 420.

a178212 n = a178212_list !! (n-1)
a178212_list = filter f [1..] where
   f x = length (a027748_row x) == 3 && any (> 1) (a124010_row x)
-- Reinhard Zumkeller, Apr 03 2015

nsD3Q[n_] := Block[{fi = FactorInteger@ n}, Length@ fi == 3 && Plus @@ Last /@ fi > 3]; Select[ Range@ 494, nsD3Q] (* Robert G. Wilson v, Feb 09 2012 *)
Select[Range[500], PrimeNu[#] == 3 && PrimeOmega[#] > 3 &] (* Alonso del Arte, Mar 23 2015, based on a comment from Robert G. Wilson v, Feb 09 2012; requires Mathematica 7.0+ *)

is_A178212(n)={ omega(n)==3 & bigomega(n)>3 }
for(n=1,999,is_A178212(n) & print1(n",")) \\ M. F. Hasler, Feb 09 2012

A001221(a(n)) = 3; A001222(a(n)) > 3; A000005(n) >= 12;

a(n) = A123712(n) for n <= 52, possibly more.

A046305 Numbers that are divisible by at least 6 primes (counted with multiplicity).

Original entry on oeis.org

64, 96, 128, 144, 160, 192, 216, 224, 240, 256, 288, 320, 324, 336, 352, 360, 384, 400, 416, 432, 448, 480, 486, 504, 512, 528, 540, 544, 560, 576, 600, 608, 624, 640, 648, 672, 704, 720, 729, 736, 756, 768, 784, 792, 800, 810, 816, 832, 840, 864, 880, 896

Offset: 1

Patrick De Geest, Jun 15 1998

nonn

John Cerkan, Table of n, a(n) for n = 1..10000

Subsequence of A033987 and A046304.

Cf. A046306.

Select[Range[1000],Total[Transpose[FactorInteger[#]][[2]]]>5&]  (* Harvey P. Dale, Jan 13 2011 *)
Select[Range[1000],PrimeOmega[#]>5&] (* Harvey P. Dale, Apr 14 2019 *)

is(n)=bigomega(n)>5 \\ Charles R Greathouse IV, Sep 17 2015

from math import prod, isqrt
from sympy import primerange, primepi, integer_nthroot
def A046305(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 almostprimepi(n, k):
        if k==0: return int(n>=1)
        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)))
        return int(sum(primepi(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
    def f(x): return n+1+sum(almostprimepi(x, k) for k in range(1, 6))
    return bisection(f, n, n) # Chai Wah Wu, Mar 29 2025

Product p_i^e_i with Sum e_i >= 6.

a(n) = n + O(n (log log n)^4/log n). - Charles R Greathouse IV, Apr 07 2017

Offset corrected by Andrew Howroyd, Aug 13 2024

A107396 a(n) = binomial(n+5, 5) * binomial(n+7, 5).

Original entry on oeis.org

21, 336, 2646, 14112, 58212, 199584, 594594, 1585584, 3864861, 8744736, 18582564, 37425024, 71954064, 132838272, 236618172, 408282336, 684723501, 1119300336, 1787771370, 2795913120, 4289184900, 6464858400, 9587091150, 14005489680, 20177780805, 28697288256

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(0+5,5)*C(0+7,5) = C(5,5)*C(7,5) = 1*21 = 21.
If n=9 then C(6+5,5)*C(6+7,5) = C(11,5)*C(13,5) = 462*1287 = 594594.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).

Cf. A033987, A062196.

A107396:= func< n | Binomial(n+5,5)*Binomial(n+7,5) >;
[A107396(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

a[n_] := Binomial[n + 5, 5] * Binomial[n + 7, 5]; Array[a, 25, 0] (* Amiram Eldar, Sep 01 2022 *)

a(n)={binomial(n+5, 5) * binomial(n+7, 5)} \\ Andrew Howroyd, Nov 08 2019

def A107396(n): return binomial(n+5,5)*binomial(n+7,5)
print([A107396(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 350*Pi^2/3 - 82901/72.
Sum_{n>=0} (-1)^n/a(n) = 1975/24 - 25*Pi^2/3. (End)
G.f.: 21*(1 + 5*x + 5*x^2 + x^3)/(1-x)^11. - G. C. Greubel, Feb 09 2025

a(7) corrected and terms a(15) and beyond from Andrew Howroyd, Nov 08 2019

A107397 a(n) = binomial(n+6, 6) * binomial(n+8, 6).

Original entry on oeis.org

28, 588, 5880, 38808, 194040, 792792, 2774772, 8588580, 24048024, 61941880, 148660512, 335785632, 719540640, 1472290848, 2891999880, 5477788008, 10042611348, 17877713700, 30988037080, 52423371000, 86736850200, 140610670200, 223698793500, 349748200620, 538074154800

Offset: 0

Zerinvary Lajos, May 25 2005

			If n=0 then C(0+6,6)*C(0+8,6) = C(6,6)*C(8,6) = 1*28 = 28.
If n=6 then C(6+6,6)*C(6+8,6) = C(12,6)*C(14,6) = 924*3003 = 2774772.

Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1).

Cf. A033987, A062196.

A107397:= func< n | Binomial(n+6,6)*Binomial(n+8,6) >;
[A107397(n): n in [0..30]]; // G. C. Greubel, Feb 09 2025

a[n_] := Binomial[n + 6, 6] * Binomial[n + 8, 6]; Array[a, 25, 0] (* Amiram Eldar, Sep 01 2022 *)

a(n)={binomial(n+6, 6) * binomial(n+8, 6)} \\ Andrew Howroyd, Nov 08 2019

def A107397(n): return binomial(n+6,6)*binomial(n+8,6)
print([A107397(n) for n in range(31)]) # G. C. Greubel, Feb 09 2025

From Amiram Eldar, Sep 01 2022: (Start)
Sum_{n>=0} 1/a(n) = 720*Pi^2 - 1740989/245.
Sum_{n>=0} (-1)^n/a(n) = 6144*log(2)/7 - 149046/245. (End)
G.f.: 28*(1 + 8*x + 15*x^2 + 8*x^3 + x^4)/(1-x)^13. - G. C. Greubel, Feb 09 2025

a(3) corrected and terms a(11) and beyond from Andrew Howroyd, Nov 08 2019

cp's OEIS Frontend

A014613 Numbers that are products of 4 primes.

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A033942 Positive integers with at least 3 prime factors (counted with multiplicity).

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A037144 Numbers with at most 3 prime factors (counted with multiplicity).

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A178254 Number of permutations of the proper divisors of n such that no adjacent elements have a common divisor greater than 1.

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A046304 Divisible by at least 5 primes (counted with multiplicity).

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A117358 a(n) = A032742(A032742(A032742(n))) = ((n/lpf(n))/lpf(n/lpf(n)))/lpf((n/lpf(n))/lpf(n/lpf(n))), where lpf=A020639, least prime factor.

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A178212 Nonsquarefree numbers divisible by exactly three distinct primes.

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