A007304 -id:A007304 - cp's OEIS frontend

A095904 Triangular array of natural numbers (greater than 1) arranged by prime signature.

2, 3, 4, 5, 9, 6, 7, 25, 10, 8, 11, 49, 14, 27, 12, 13, 121, 15, 125, 18, 16, 17, 169, 21, 343, 20, 81, 24, 19, 289, 22, 1331, 28, 625, 40, 30, 23, 361, 26, 2197, 44, 2401, 54, 42, 32, 29, 529, 33, 4913, 45, 14641, 56, 66, 243, 36, 31, 841, 34, 6859, 50, 28561, 88, 70

Offset: 0

Alford Arnold, Jul 10 2004

The unit, 1, has the empty prime signature { } (thus not in triangle).
Downwards diagonals:
* Rightmost diagonal: smallest numbers of a given prime signature in increasing order (A025487). This defines the order of signatures used.
  This special ordering of prime signatures (by increasing smallest numbers of a given prime signature, A181087) is unrelated to any of the 8 variants of graded lexicographic or colexicographic orderings (based on the exponents only) since it depends on the magnitudes of the prime numbers. It is not even graded by Omega(n).
* Second rightmost diagonal: second smallest numbers of a given prime signature (A077560). (They are not increasing anymore.)
Upwards diagonals:
* Leftmost diagonal: primes.                           {1}     (A000040)
* 2nd leftmost diagonal: squares of primes.            {2}     (A001248)
* 3rd leftmost diagonal: squarefree biprimes.          {1,1}   (A006881)
* 4th leftmost diagonal: cubes of primes.              {3}     (A030078)
* 5th leftmost diagonal: signature (Achilles numbers)  {1,2}   (A054753)
* 6th leftmost diagonal: fourth powers of primes.      {4}     (A030514)
* 7th leftmost diagonal: signature (Achilles numbers)  {1,3}   (A065036)
* 8th leftmost diagonal: squarefree triprimes.         {1,1,1} (A007304)
  The Achilles numbers are nonsquarefree while not perfect powers.
Prime signatures are often expressed in increasing order of exponents. The decreasing order of exponents (as on the Wiki page, see links) has the advantage of listing the exponents in the same order (with the canonical factorization convention) as the smallest number of a given prime signature.

			343 is in the 4th left- and 4th rightmost diagonal, because it is the 4th value with the 4th prime signature {3}.
First 8 rows of triangular array (Cf. table link for this sequence):
                                   2
                              3         4
                         5         9         6
                    7        25        10         8
               11       49        14        27        12
          13      121        15       125        18        16
     17       169       21       343        20        81        24
19       289       22       1331       28       625        40        30

OEIS Wiki, Prime signature.
OEIS Wiki, Orderings.

Cf. A083140, A064839, A181087.

Extended by Ray Chandler, Jul 31 2004
Corrected (minor) by Daniel Forgues, Jan 21 2011
Example, comments by Daniel Forgues, Jan 21 2011
Edited by Alois P. Heinz, Jan 23 2011
Edited by Daniel Forgues, Jan 23 2011

A123321 Products of 7 distinct primes (squarefree 7-almost primes).

Original entry on oeis.org

510510, 570570, 690690, 746130, 870870, 881790, 903210, 930930, 1009470, 1067430, 1111110, 1138830, 1193010, 1217370, 1231230, 1272810, 1291290, 1345890, 1360590, 1385670, 1411410, 1438710, 1452990, 1504230, 1540770

Offset: 1

Rick L. Shepherd, Sep 25 2006

nonn

Intersection of A005117 and A046308.

Intersection of A005117 and A176655. - R. J. Mathar, Dec 05 2016

			a(1) = 510510 = 2*3*5*7*11*13*17 = A002110(7).

Rick L. Shepherd, Table of n, a(n) for n = 1..10000

Cf. A005117, A046308, A048692, Squarefree k-almost primes: A000040 (k=1), A006881 (k=2), A007304 (k=3), A046386 (k=4), A046387 (k=5), A067885 (k=6), A123322 (k=8), A115343 (k=9).

f7Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1, 1, 1}; lst={};Do[If[f7Q[n], AppendTo[lst, n]], {n, 9!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Select[Range[1600000],PrimeNu[#]==7&&SquareFreeQ[#]&] (* Harvey P. Dale, Sep 19 2013 *)

is(n)=omega(n)==7 && bigomega(n)==7 \\ Hugo Pfoertner, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A123321(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,7)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

More terms from Vladimir Joseph Stephan Orlovsky, Aug 26 2008

A337453 Numbers k such that the k-th composition in standard order is an ordered triple of distinct positive integers.

Original entry on oeis.org

37, 38, 41, 44, 50, 52, 69, 70, 81, 88, 98, 104, 133, 134, 137, 140, 145, 152, 161, 176, 194, 196, 200, 208, 261, 262, 265, 268, 274, 276, 289, 290, 296, 304, 321, 324, 328, 352, 386, 388, 400, 416, 517, 518, 521, 524, 529, 530, 532, 536, 545, 560, 577, 578

Offset: 1

Gus Wiseman, Sep 07 2020

nonn

The k-th composition in standard order (graded reverse-lexicographic, A066099) is obtained by taking the set of positions of 1's in the reversed binary expansion of k, prepending 0, taking first differences, and reversing again. This gives a bijective correspondence between nonnegative integers and integer compositions.

			The sequence together with the corresponding triples begins:
     37: (3,2,1)    140: (4,1,3)    289: (3,5,1)
     38: (3,1,2)    145: (3,4,1)    290: (3,4,2)
     41: (2,3,1)    152: (3,1,4)    296: (3,2,4)
     44: (2,1,3)    161: (2,5,1)    304: (3,1,5)
     50: (1,3,2)    176: (2,1,5)    321: (2,6,1)
     52: (1,2,3)    194: (1,5,2)    324: (2,4,3)
     69: (4,2,1)    196: (1,4,3)    328: (2,3,4)
     70: (4,1,2)    200: (1,3,4)    352: (2,1,6)
     81: (2,4,1)    208: (1,2,5)    386: (1,6,2)
     88: (2,1,4)    261: (6,2,1)    388: (1,5,3)
     98: (1,4,2)    262: (6,1,2)    400: (1,3,5)
    104: (1,2,4)    265: (5,3,1)    416: (1,2,6)
    133: (5,2,1)    268: (5,1,3)    517: (7,2,1)
    134: (5,1,2)    274: (4,3,2)    518: (7,1,2)
    137: (4,3,1)    276: (4,2,3)    521: (6,3,1)

6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1) counts these compositions.
A007304 is an unordered version.
A014311 is the non-strict version.
A337461 counts the coprime case.
A000217(n - 2) counts 3-part compositions.
A001399(n - 3) = A069905(n) = A211540(n + 2) counts 3-part partitions.
A001399(n - 6) = A069905(n - 3) = A211540(n - 1) counts strict 3-part partitions.
A014612 ranks 3-part partitions.
Cf. A000212, A220377, A307534, A337459, A337460, A337561, A337603, A337604.

stc[n_]:=Differences[Prepend[Join@@Position[Reverse[IntegerDigits[n,2]],1],0]]//Reverse;
Select[Range[0,100],Length[stc[#]]==3&&UnsameQ@@stc[#]&]

These triples are counted by 6*A001399(n - 6) = 6*A069905(n - 3) = 6*A211540(n - 1).

Intersection of A014311 and A233564.

A067813 Start of a record-breaking run of consecutive integers with a number of prime factors (counted with multiplicity) equal to 3.

Original entry on oeis.org

8, 27, 170, 602, 2522, 211673

Offset: 1

Shyam Sunder Gupta, Feb 07 2002

602 is the first number having 4 and 5 consecutive integers with 3 prime factors. - T. D. Noe, Mar 19 2014

			a(4)=602 because 602 is the start of a record breaking run of 5 consecutive integers (602 to 606) each having 3 prime factors; i.e. bigomega(n)=A001222(n)=3 for n = 602, ..., 606.

Cf. A007304, A045984, A067814, A067820, A067821, A067822, A117969, A259756.

bigomega[n_] := Plus@@Last/@FactorInteger[n]; For[n=1; m=l=0, True, n++, If[bigomega[n]==3, l++, If[l>m, m=l; Print[n-l, " ", l]]; l=0]]
Module[{nn=8,po},po=PrimeOmega[Range[5000000]];Flatten[Table[ SequencePosition[ po,PadRight[{},n,3],1],{n,nn}],1]][[All,1]]//Union (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Jun 14 2019 *)

show(lim)=my(was,r,ct); forfactored(n=2, lim\1+1, is=vecsum(n[2][, 2])==3; if(is, ct++; if(ct>r, r=ct; print(r" "n[1]-r+1)),ct=0)) \\ Charles R Greathouse IV, Jun 26 2019

Edited by Dean Hickerson, Jul 31 2002

A162143 Numbers that are the squares of the product of three distinct primes.

Original entry on oeis.org

900, 1764, 4356, 4900, 6084, 10404, 11025, 12100, 12996, 16900, 19044, 23716, 27225, 28900, 30276, 33124, 34596, 36100, 38025, 49284, 52900, 53361, 56644, 60516, 65025, 66564, 70756, 74529, 79524, 81225, 81796, 84100, 96100, 101124, 103684, 119025, 125316, 127449

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

nonn

Numbers that are the product of exactly 3 distinct squares of primes (p^2*q^2*r^2).

			900 = 2^2*3^2*5^2, 1764 = 2^2*3^2*7^2, 4356 = 2^2*3^2*11^2, ..

Cf. A007304, A006881, A050326, A162142, A085548, A085964, A085966.

h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
A162143List := upto -> seq(n^2, n=select(h, [seq(1..upto)])):  # Peter Luschny, Apr 14 2025

fQ[n_]:=Last/@FactorInteger[n]=={2,2,2}; Select[Range[100000], f]

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162143(n):
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f)**2 # Chai Wah Wu, Aug 29 2024

def is_a(n):
    P = prime_divisors(n)
    return len(P) == 3 and prod(P) == n
print([n*n for n in range(1, 439) if is_a(n)]) # Peter Luschny, Apr 14 2025

a(n) = A007304(n)^2.
A050326(a(n)) = 8. - Reinhard Zumkeller, May 03 2013
Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 = (A085548^3 + 2*A085966 - 3*A085548*A085964)/6 = 0.0036962441..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Edited by N. J. A. Sloane, Jun 27 2009

A179668 Products of the 8th power of a prime and a distinct prime (p^8*q).

Original entry on oeis.org

768, 1280, 1792, 2816, 3328, 4352, 4864, 5888, 7424, 7936, 9472, 10496, 11008, 12032, 13122, 13568, 15104, 15616, 17152, 18176, 18688, 20224, 21248, 22784, 24832, 25856, 26368, 27392, 27904, 28928, 32512, 32805, 33536, 35072, 35584, 38144, 38656, 40192

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 23 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,8}; Select[Range[40000], f]
With[{nn=40},Take[Union[#[[1]]^8 #[[2]]&/@Flatten[Permutations/@Subsets[ Prime[Range[nn]],{2}],1]],nn]] (* Harvey P. Dale, Jan 20 2016 *)

list(lim)=my(v=List(),t);forprime(p=2,(lim\2)^(1/8),t=p^8;forprime(q=2,lim\t,if(p==q,next);listput(v,t*q)));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 20 2011

from sympy import primepi, primerange, integer_nthroot
def A179668(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(primepi(x//p**8) for p in primerange(integer_nthroot(x,8)[0]+1))+primepi(integer_nthroot(x,9)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A179692 Numbers of the form p^9*q where p and q are distinct primes.

Original entry on oeis.org

1536, 2560, 3584, 5632, 6656, 8704, 9728, 11776, 14848, 15872, 18944, 20992, 22016, 24064, 27136, 30208, 31232, 34304, 36352, 37376, 39366, 40448, 42496, 45568, 49664, 51712, 52736, 54784, 55808, 57856, 65024, 67072, 70144, 71168, 76288, 77312, 80384, 83456

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jul 24 2010

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000
Will Nicholes, Prime Signatures

Cf. A006881, A007304, A065036, A085986, A085987, A092759, A178739, A179642, A179643, A179644, A179645, A179646, A179664, A179665, A179666, A179667, A179668, A179669, A179670, A179671, A179672, A179688, A179689, A179690, A179691.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,9}; Select[Range[90000], f]

list(lim)=my(v=List(),t);forprime(p=2, (lim\2)^(1/9), t=p^9;forprime(q=2, lim\t, if(p==q, next);listput(v,t*q))); vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 24 2011

from sympy import primepi, integer_nthroot, primerange
def A179692(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(primepi(x//p**9) for p in primerange(integer_nthroot(x,9)[0]+1))+primepi(integer_nthroot(x,10)[0])
    return bisection(f,n,n) # Chai Wah Wu, Feb 21 2025

A277120 Number of branching factorizations of n.

Original entry on oeis.org

0, 1, 1, 2, 1, 3, 1, 5, 2, 3, 1, 11, 1, 3, 3, 15, 1, 11, 1, 11, 3, 3, 1, 45, 2, 3, 5, 11, 1, 19, 1, 51, 3, 3, 3, 62, 1, 3, 3, 45, 1, 19, 1, 11, 11, 3, 1, 195, 2, 11, 3, 11, 1, 45, 3, 45, 3, 3, 1, 113, 1, 3, 11, 188, 3, 19, 1, 11, 3, 19, 1, 345, 1, 3, 11, 11, 3

Offset: 1

Michel Marcus, Oct 01 2016

nonn

Per the formula, a(n) = 1 at prime n. As the sequence extends, additional values become more frequent than 1. These values can be characterized, for example, a(n) = 19 is seen at n corresponding to A007304, a(n) = 3 is seen at n corresponding to A006881, a(n) = 113 is seen at n corresponding to A085987. - Bill McEachen, Dec 28 2023
From Antti Karttunen, Jan 02 2024: (Start)
The value of a(n) depends only on the prime signature of n. In other words, for all i, j >= 1, it holds that A101296(i) = A101296(j) => a(i) = a(j). Moreover, it seems that the converse proposition also holds, that for all i, j >= 1, a(i) = a(j) => A101296(i) = A101296(j), i.e., for each distinct prime signature there exists a distinct value of a(n). This has been empirically checked up to the first 21001 prime signatures in A025487 (see A366884), and can be proved if one can show that the latter sequence (equally: A366377) is injective. If this conjecture holds, it would imply an unlimited number of statements like those given in the previous comment (see the formula section of A101296).
Questions: Are there any terms of the form 10k+4 or 10k+6? What is the asymptotic density of terms of the form 10k+5 (those ending with digit "5")? Compare to the data shown in A366884.
For squarefree n > 1, a(n) is never even, and apparently, never a multiple of five. See comments in A052886.
(End)

			In this scheme, the following factorizations of 12 are counted as distinct: 12, 2 x 6, 2 x (2 x 3), 2 x (3 x 2), 3 x 4, 3 x (2 x 2), 4 x 3, (2 x 2) x 3, 6 x 2, (2 x 3) x 2, (3 x 2) x 2, thus a(12) = 11. - _Antti Karttunen_, Nov 02 2016, based on the illustration given at page 14 of Knopfmacher & Mays paper.
The following factorizations of 30 are counted as distinct: 30, 2 x 15, 15 x 2, 3 x 10, 10 x 3, 5 x 6, 6 x 5, 2 x (3 x 5), 2 x (5 x 3), 3 x (2 x 5), 3 x (5 x 2), 5 x (2 x 3), 5 x (3 x 2), (2 x 3) x 5, (2 x 5) x 3, (3 x 2) x 5, (3 x 5) x 2, (5 x 2) x 3, (5 x 3) x 2, thus a(30) = 19. - _Antti Karttunen_, Jan 02 2024

Daniel Mondot, Table of n, a(n) for n = 1..9999
H.-K. Hwang, Théorèmes limites pour les structures combinatoires et les fonctions arithmétiques, PhD thesis, in Ecole Polytechnique, 1994. See page 198.
A. Knopfmacher, M. E. Mays, A survey of factorization counting functions, International Journal of Number Theory, 1(4):563-581,(2005). See page 14.
Index entries for sequences computed from exponents in factorization of n

Cf. A007317, A052886, A074206, A101296, A277130, A366377 [= a(A108951(n))], A366884 [= a(A025487(n))].

After n=1 differs from A104725 for the next time at n=32, where a(32) = 51, while A104725(32) = 52.

#include 
#define MAX 10000
/* Number of branching factorizations of n. */
unsigned long n, m, a, b, p, x, nbr[MAX];
int main(void)
{
  for (x=n=1; nDaniel Mondot, Oct 01 2016 */

v[n_] := v[n] = If[n == 1, 0, 1 + Sum[If[d == 1 || d^2 > n, 0, If[d^2 == n, 1, 2]*v[d]*v[n/d]], {d, Divisors[n]}]]; Table[v[n], {n, 1, 100}] (* Vaclav Kotesovec, Jan 13 2024, after Antti Karttunen *)

A277120(n) = if(1==n, 0, 1+sumdiv(n, d, if((1==d)||(d*d)>n,0,if((d*d)==n,1,2)*A277120(d)*A277120(n/d)))); \\ Antti Karttunen, Nov 02 2016, after Daniel Mondot's C-program above.

seq(n)={my(v=vector(n)); for(n=2, n, v[n] = 1 + sumdiv(n, d, v[d]*v[n/d])); v} \\ Andrew Howroyd, Nov 17 2018

a(1) = 0; for n > 1, a(n) = 1 + Sum_{d|n, 1 < d < n} a(d)*a(n/d). - Antti Karttunen, Nov 02 2016, after Daniel Mondot's C program, simplified Dec 30 2023.

For all n >= 1, a(prime^n) = A007317(n), and a(product of n distinct primes) = A052886(n). - Antti Karttunen, Dec 31 2023

More terms from Daniel Mondot, Oct 01 2016

A100565 a(n) = Card{(x,y,z) : x <= y <= z, x|n, y|n, z|n, gcd(x,y)=1, gcd(x,z)=1, gcd(y,z)=1}.

Original entry on oeis.org

1, 2, 2, 3, 2, 5, 2, 4, 3, 5, 2, 8, 2, 5, 5, 5, 2, 8, 2, 8, 5, 5, 2, 11, 3, 5, 4, 8, 2, 15, 2, 6, 5, 5, 5, 13, 2, 5, 5, 11, 2, 15, 2, 8, 8, 5, 2, 14, 3, 8, 5, 8, 2, 11, 5, 11, 5, 5, 2, 25, 2, 5, 8, 7, 5, 15, 2, 8, 5, 15, 2, 18, 2, 5, 8, 8, 5, 15, 2, 14, 5, 5, 2, 25, 5, 5, 5, 11, 2, 25, 5, 8, 5, 5, 5, 17

Offset: 1

Vladeta Jovovic, Nov 28 2004

First differs from A018892 at a(30) = 15, A018892(30) = 14.
First differs from A343654 at a(210) = 51, A343654(210) = 52.
Also a(n) = Card{(x,y,z) : x <= y <= z and lcm(x,y)=n, lcm(x,z)=n, lcm(y,z)=n}.
In words, a(n) is the number of pairwise coprime unordered triples of divisors of n. - Gus Wiseman, May 01 2021

			From _Gus Wiseman_, May 01 2021: (Start)
The a(n) triples for n = 1, 2, 4, 6, 8, 12, 24:
  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)  (1,1,1)   (1,1,1)
           (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)  (1,1,2)   (1,1,2)
                    (1,1,4)  (1,1,3)  (1,1,4)  (1,1,3)   (1,1,3)
                             (1,1,6)  (1,1,8)  (1,1,4)   (1,1,4)
                             (1,2,3)           (1,1,6)   (1,1,6)
                                               (1,2,3)   (1,1,8)
                                               (1,3,4)   (1,2,3)
                                               (1,1,12)  (1,3,4)
                                                         (1,3,8)
                                                         (1,1,12)
                                                         (1,1,24)
(End)

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A070919, A086222, A086165, A048691, A063647.
Positions of 2's through 5's are A000040, A001248, A030078, A068993.
The version for subsets of {1..n} instead of divisors is A015617.
The version for pairs of divisors is A018892.
The ordered version is A048785.
The strict case is A066620.
The version for strict partitions is A220377.
A version for sets of divisors of any size is A225520.
The version for partitions is A307719 (no 1's: A337563).
The case of distinct parts coprime is A337600 (ordered: A337602).
A001399(n-3) = A069905(n) = A211540(n+2) counts 3-part partitions.
A007304 ranks 3-part strict partitions.
A014311 ranks 3-part compositions.
A014612 ranks 3-part partitions.
A051026 counts pairwise indivisible subsets of {1..n}.
A302696 lists Heinz numbers of pairwise coprime partitions.
A337461 counts 3-part pairwise coprime compositions.
Cf. A000961, A000977, A007360, A023022, A087087, A276187, A282935, A337601, A337603, A338331, A343652, A343654.

pwcop[y_]:=And@@(GCD@@#==1&/@Subsets[y,{2}]);
Table[Length[Select[Tuples[Divisors[n],3],LessEqual@@#&&pwcop[#]&]],{n,30}] (* Gus Wiseman, May 01 2021 *)

A100565(n) = (numdiv(n^3)+3*numdiv(n)+2)/6; \\ Antti Karttunen, May 19 2017

a(n) = (tau(n^3) + 3*tau(n) + 2)/6.

A115343 Products of 9 distinct primes.

Original entry on oeis.org

223092870, 281291010, 300690390, 340510170, 358888530, 363993630, 380570190, 397687290, 406816410, 417086670, 434444010, 455885430, 458948490, 481410930, 485555070, 497668710, 504894390, 512942430, 514083570, 531990690, 538047510, 547777230, 551861310

Offset: 1

Jonathan Vos Post, Mar 06 2006

			514083570 is in the sequence as it is equal to 2*3*5*7*11*13*17*19*53.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1045 terms from Vincenzo Librandi and Chai Wah Wu)

Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

N:= 10^9: # to get all terms < N
n0:= mul(ithprime(i),i=1..8):
Primes:= select(isprime,[$1..floor(N/n0)]):
nPrimes:= nops(Primes):
for i from 1 to 9 do
  for j from 1 to nPrimes do
    M[i,j]:= convert(Primes[1..min(j,i)],`*`);
od od:
A:= {}:
for i9 from 9 to nPrimes do
  m9:= Primes[i9];
for i8 in select(t -> M[7,t-1]*Primes[t]*m9 <= N, [$8..i9-1]) do
  m8:= m9*Primes[i8];
for i7 in select(t -> M[6,t-1]*Primes[t]*m8 <= N, [$7..i8-1]) do
  m7:= m8*Primes[i7];
for i6 in select(t -> M[5,t-1]*Primes[t]*m7 <= N, [$6..i7-1]) do
  m6:= m7*Primes[i6];
for i5 in select(t -> M[4,t-1]*Primes[t]*m6 <= N, [$5..i6-1]) do
  m5:= m6*Primes[i5];
for i4 in select(t -> M[3,t-1]*Primes[t]*m5 <= N, [$4..i5-1]) do
  m4:= m5*Primes[i4];
for i3 in select(t -> M[2,t-1]*Primes[t]*m4 <= N, [$3..i4-1]) do
  m3:= m4*Primes[i3];
for i2 in select(t -> M[1,t-1]*Primes[t]*m3 <= N, [$2..i3-1]) do
  m2:= m3*Primes[i2];
for i1 in select(t -> Primes[t]*m2 <= N, [$1..i2-1]) do
  A:= A union {m2*Primes[i1]};
od od od od od od od od od:
A; # Robert Israel, Sep 02 2014

Module[{n=6*10^8,k},k=PrimePi[n/Times@@Prime[Range[8]]];Select[ Union[ Times@@@ Subsets[Prime[Range[k]],{9}]],#<=n&]](* Harvey P. Dale with suggestions from Jean-François Alcover, Sep 03 2014 *)
n = 10^9; n0 = Times @@ Prime[Range[8]]; primes = Select[Range[Floor[n/n0]], PrimeQ]; nPrimes = Length[primes]; Do[M[i, j] = Times @@ primes[[1 ;; Min[j, i]]], {i, 1, 9}, {j, 1, nPrimes}]; A = {};
Do[m9 = primes[[i9]];
Do[m8 = m9*primes[[i8]];
Do[m7 = m8*primes[[i7]];
Do[m6 = m7*primes[[i6]];
Do[m5 = m6*primes[[i5]];
Do[m4 = m5*primes[[i4]];
Do[m3 = m4*primes[[i3]];
Do[m2 = m3*primes[[i2]];
Do[A = A ~Union~ {m2*primes[[i1]]},
{i1, Select[Range[1, i2-1], primes[[#]]*m2 <= n &]}],
{i2, Select[Range[2, i3-1], M[1, #-1]*primes[[#]]*m3 <= n &]}],
{i3, Select[Range[3, i4-1], M[2, #-1]*primes[[#]]*m4 <= n &]}],
{i4, Select[Range[4, i5-1], M[3, #-1]*primes[[#]]*m5 <= n &]}],
{i5, Select[Range[5, i6-1], M[4, #-1]*primes[[#]]*m6 <= n &]}],
{i6, Select[Range[6, i7-1], M[5, #-1]*primes[[#]]*m7 <= n &]}],
{i7, Select[Range[7, i8-1], M[6, #-1]*primes[[#]]*m8 <= n &]}],
{i8, Select[Range[8, i9-1], M[7, #-1]*primes[[#]]*m9 <= n &]}],
{i9, 9, nPrimes}];
A (* Jean-François Alcover, Sep 03 2014, translated and adapted from Robert Israel's Maple program *)

is(n)=omega(n)==9 && bigomega(n)==9 \\ Hugo Pfoertner, Dec 18 2018

from operator import mul
from functools import reduce
from sympy import nextprime, sieve
from itertools import combinations
n = 190
m = 9699690*nextprime(n-1)
A115343 = []
for x in combinations(sieve.primerange(1,n),9):
    y = reduce(mul,(d for d in x))
    if y < m:
        A115343.append(y)
A115343 = sorted(A115343) # Chai Wah Wu, Sep 02 2014

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A115343(n):
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) for d in g(x,a2,b2,c*b2,m-1)))
    def f(x): return int(n+x-sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,9)))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return bisection(f) # Chai Wah Wu, Aug 31 2024

Corrected and extended by Don Reble, Mar 09 2006

More terms and corrected b-file from Chai Wah Wu, Sep 02 2014

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