A067885 -id:A067885 - cp's OEIS frontend

A340467 a(n) is the n-th squarefree number having n prime factors.

2, 10, 66, 462, 4290, 53130, 903210, 17687670, 406816410, 11125544430, 338431883790, 11833068917670, 457077357006270, 20384767656323070, 955041577211912190, 49230430891074322890, 2740956243836856315270, 168909608387276001835590, 11054926927790884163355330

Offset: 1

Alois P. Heinz, Jan 08 2021

nonn

a(n) is the n-th product of n distinct primes.
All terms are even.
This sequence differs from A073329 which has also nonsquarefree terms.

			a(1) = A000040(1) = 2.
a(2) = A006881(2) = 10.
a(3) = A007304(3) = 66.
a(4) = A046386(4) = 462.
a(5) = A046387(5) = 4290.
a(6) = A067885(6) = 53130.
a(7) = A123321(7) = 903210.
a(8) = A123322(8) = 17687670.
a(9) = A115343(9) = 406816410.
a(10) = A281222(10) = 11125544430.

Alois P. Heinz, Table of n, a(n) for n = 1..350

Main diagonal of A340316.
Cf. A001221, A001222, A005117, A070826, A072047, A073329, A101695, A340313.
Cf. A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A340467(n):
    if n == 1: return 2
    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,n)))
    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

a(n) = A340316(n,n).
a(n) = A005117(m) <=> A072047(m) = n = A340313(m).
A001221(a(n)) = A001222(a(n)) = n.
a(n) < A070826(n+1), the least odd number with exactly n distinct prime divisors.

A346014 Numbers whose average number of distinct prime factors of their divisors is an integer.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 106, 111, 115, 118, 119, 122, 123, 129, 133, 134, 141, 142, 143, 145, 146, 155, 158, 159, 161, 166, 177, 178, 183, 185, 187, 194, 201, 202

Offset: 1

Amiram Eldar, Jul 01 2021

nonn

First differs from A030229 at n = 275. a(275) = 900 is the least term that is not squarefree and therefore not in A030229.
The least term whose exponents in its prime factorization are not all the same is 1080 = 2^3 * 3^3 * 5.
The least term whose exponents in its prime factorization are distinct is 1440 = 2^5 * 3^2 * 5.
Numbers k such that A000005(k) | A062799(k).
Numbers k such that A346010(k) = 1.
Numbers k such that if the prime factorization of k is Product_{i} p_i^e_i, then Sum_{i} e_i/(e_i + 1) is an integer.
Includes all the squarefree numbers with an even number of prime divisors (A030229), i.e., the union of A006881, A046386, A067885, A123322, ...
If k is squarefree with m prime divisors then k^(m-1) is a term. E.g., the squares of the sphenic numbers (A162143) are terms.

			6 is a term since it has 4 divisors, 1, 2, 3 and 6 and (omega(1) + omega(2) + omega(3) + omega(6))/4 = (0 + 1 + 1 + 2)/4 = 1 is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A001221, A062799, A346009, A346010.

Subsequences: A006881, A030229, A046386, A067885, A123322, A162143.

f[p_, e_] := e/(e + 1); d[1] = 1; d[n_] := Denominator[Plus @@ f @@@ FactorInteger[n]]; Select[Range[200], d[#] == 1 &]

A214195 Numbers with the number of distinct prime factors a multiple of 3.

Original entry on oeis.org

1, 30, 42, 60, 66, 70, 78, 84, 90, 102, 105, 110, 114, 120, 126, 130, 132, 138, 140, 150, 154, 156, 165, 168, 170, 174, 180, 182, 186, 190, 195, 198, 204, 220, 222, 228, 230, 231, 234, 238, 240, 246, 252, 255, 258, 260, 264, 266, 270, 273, 276, 280, 282, 285

Offset: 1

Enrique Pérez Herrero, Jul 07 2012

nonn

If GCD(a(n),a(m))=1, then a(n)*a(m) is also in this sequence. - Enrique Pérez Herrero, Nov 23 2013

Enrique Pérez Herrero, Table of n, a(n) for n = 1..5000

Subsequences include A033992, A067885, A007304 and A147573.

Cf. A145784, A030230, A030231.

Select[Range[1000],Mod[PrimeNu[#],3]==0&]

is(n)=omega(n)%3==0 \\ Charles R Greathouse IV, Sep 14 2015

A010872(A001221(a(n))) = 0.

A378097 Products of 6 distinct primes that are sandwiched between twin prime numbers.

Original entry on oeis.org

43890, 51870, 84630, 102102, 140070, 149730, 153510, 168630, 224070, 251790, 269178, 281190, 308490, 316470, 317730, 322770, 355110, 376530, 381990, 383838, 389298, 404430, 432390, 434010, 459030, 467670, 486330, 487830, 496230, 506730, 520410, 531570, 545790, 552090, 560490, 573342, 576030, 583338

Offset: 1

Massimo Kofler, Nov 16 2024

nonn

All the terms are divisible by 6.

			43890 is in the sequence a term because 43890=2*3*5*7*11*19 is the product of six distinct primes and 43889, 43891 are a couple of twin primes.
51870 is in the sequence a term because 51870=2*3*5*7*13*19 is the product of six distinct primes and 51869, 51871 are a couple of twin primes.

Intersection of A014574 and A067885.

Cf. A083207 (supersequence), A353022, A376380.

Select[6 * Range[10^5], PrimeQ[#-1] && PrimeQ[#+1] && FactorInteger[#][[;;, 2]] == {1,1,1,1,1,1} &] (* Amiram Eldar, Nov 16 2024 *)

A378627 Products of 6 distinct primes that are sandwiched between semiprime numbers.

Original entry on oeis.org

39270, 66990, 71610, 79170, 82110, 99330, 110670, 122430, 123690, 125970, 129030, 132090, 136290, 144690, 152490, 163590, 166530, 167790, 180642, 182910, 190190, 191730, 215670, 220110, 222222, 226590, 227766, 231990, 235410, 239190, 247170, 248710, 249690, 254562, 258258, 260130

Offset: 1

Massimo Kofler, Dec 02 2024

nonn

All terms are even.

Not all terms are divisible by 6: the first that is not is a(21) = 190190. The first term that is deficient is a(1966) = 4739702. - Robert Israel, Feb 03 2025

			39270 is a term because 39270=2*3*5*7*11*17 is the product of six distinct primes, 39269=107*367 and 39271=173*227 are both semiprimes.
66990 is a term because 66990=2*3*5*7*11*29 is the product of six distinct primes, 66989=13*5153 and 66991=31*2161 are both semiprimes.

Robert Israel, Table of n, a(n) for n = 1..10000

Intersection of A067885 and A124936.

Cf. A001358, A378097.

with(priqueue):
children:= proc(t) local R,i,pp;
   R:= NULL:
   pp:= nextprime(t[6]);
   for i from 6 to 2 by -1 do
     R:= R, [t[1]*pp/t[i], op(t[2..i-1]),op(t[i+1..6]),pp];
     if t[i-1] <> prevprime(t[i]) then break fi;
   od;
   {R}
end proc:
Res:= NULL: count:= 0:
initialize(pq):
insert([-2*mul(ithprime(i),i=2..6),3,5,7,11,13],pq);
while count < 100 do
  t:= extract(pq);
  if numtheory:-bigomega(-t[1]-1) = 2 and numtheory:-bigomega(-t[1]+1) = 2 then
    Res:= Res, -t[1]; count:= count+1;
  fi;
  for tt in children(t) do insert(tt,pq) od:
od:
Res; # Robert Israel, Feb 03 2025

SequencePosition[Array[FactorInteger[#][[;; , 2]] &, 270000] /. {2} -> {1, 1}, {{1, 1}, {1, 1, 1, 1, 1, 1}, {1, 1}}][[;; , 1]] + 1 (* Amiram Eldar, Dec 02 2024 *)

Edited by Robert Israel, Feb 03 2025

A172443 Numbers with exactly 64 divisors.

Original entry on oeis.org

7560, 9240, 10920, 11880, 13440, 14040, 14280, 15960, 16632, 17160, 17280, 18360, 19320, 19656, 20520, 20790, 21000, 21120, 22440, 24024, 24192, 24360, 24570, 24840, 24960, 25080, 25704, 26040, 26520, 27000, 28728, 29568, 29640, 30030, 30360, 30888, 31080

Offset: 1

Harvey P. Dale, Nov 20 2010

The first squarefree term of this sequence is the primorial a(34) = 30030.

Almost all terms of this sequence (in the sense of having relative density 1) are squarefree, that is in our case, the product of six distinct primes = A067885. - Charles R Greathouse IV, Aug 27 2021

			10920 has 64 divisors.

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A067885.

Select[Range[100000],DivisorSigma[0,#]==64&]

is(n) = numdiv(n) == 64 \\ David A. Corneth, Aug 27 2021

from sympy import divisor_count
def ok(n): return divisor_count(n) == 64
print(list(filter(ok, range(31100)))) # Michael S. Branicky, Aug 27 2021

A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

Original entry on oeis.org

3, 5, 27, 7, 10, 11, 9, 25, 14, 13, 20, 17, 15, 21, 7625597484987, 19, 24, 23, 28, 22, 26, 29, 50, 32, 33, 3125, 44, 31, 42, 37, 49, 34, 35, 38, 100, 41, 39, 46, 45, 43, 66, 47, 52, 56, 51, 53, 80, 121, 98, 55, 54, 59, 68, 57, 63, 58, 62, 61, 84, 67, 65, 75

Offset: 2

Rémy Sigrist, Feb 07 2017

nonn

The prime tower factorization of a number is defined in A182318.
The prime tower factorization equivalence classes are described in A279686.
For any n>1, a(n)=least k>n such that A279690(n)=A279690(k).
This sequence is a permutation of the complement of A279686.
This sequence is to prime tower factorization what A081761 is to prime signature.

Rémy Sigrist, Table of n, a(n) for n = 2..2500
Rémy Sigrist, Illustration of the first terms

Cf. A000040, A006881, A007304, A046386, A046387, A051674, A067885, A081761, A182318, A279686, A279690

a(n) = my (c=a279690(n)); my (k=n+1); while (c!=a279690(k), k++); k

a(A000040(n)) = A000040(n+1) for any n>0.
a(A006881(n)) = A006881(n+1) for any n>0.
a(A051674(n)) = A051674(n+1) for any n>0.
a(A007304(n)) = A007304(n+1) for any n>0.
a(A046386(n)) = A046386(n+1) for any n>0.
a(A046387(n)) = A046387(n+1) for any n>0.
a(A067885(n)) = A067885(n+1) for any n>0.

A327829 Squarefree numbers with a prime number of prime factors.

Original entry on oeis.org

6, 10, 14, 15, 21, 22, 26, 30, 33, 34, 35, 38, 39, 42, 46, 51, 55, 57, 58, 62, 65, 66, 69, 70, 74, 77, 78, 82, 85, 86, 87, 91, 93, 94, 95, 102, 105, 106, 110, 111, 114, 115, 118, 119, 122, 123, 129, 130, 133, 134, 138, 141, 142, 143, 145, 146, 154, 155, 158, 159, 161

Offset: 1

Sebastian F. Orellana, Sep 26 2019

nonn

210 is the first integer in A120944 but not here: it has 4 prime factors.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A120944.
A006881, A007304, A046387 are subsequences.
A046386, A067885 are not subsequences.

Select[Range@ 161, And[SquareFreeQ@ #, PrimeQ@ PrimeNu@ #] &] (* Michael De Vlieger, Sep 29 2019 *)

isok(n) = issquarefree(n) && isprime(omega(n)); \\ Michel Marcus, Sep 27 2019

A006881 UNION A007304 UNION A046387 UNION A123321 UNION .... - R. J. Mathar, Oct 13 2019

Corrected and extended by Michel Marcus, Sep 27 2019

A356683 a(n) is the smallest positive k such that the count of squarefree numbers <= k that have n prime factors is equal to the count of squarefree numbers <= k that have n-1 prime factors (and the count is positive).

Original entry on oeis.org

2, 39, 1279786, 8377774397163159586

Offset: 1

Jon E. Schoenfield, Nov 22 2022

			The first two squarefree numbers are 1 and 2; 1 has 0 prime factors and 2 has 1 prime factor, so a(1)=2.
At k=39, in the interval [1..k], there are 12 squarefree numbers with 1 prime factor (i.e., 12 primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37), and 12 squarefree numbers with 2 prime factors (i.e., 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39). k=39 is the smallest such positive number for which these two counts are the same (and are positive), so a(2)=39.
At k=1279786, the interval [1..k] includes 265549 squarefree numbers with 2 prime factors and the same number of squarefree numbers with 3 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(3)=1279786. There are 75 numbers with this property, the last one being 1281378.
At k=8377774397163159586, the interval [1..k] includes 1356557942402075858 squarefree numbers with 3 prime factors and the same number of squarefree numbers with 4 prime factors, and there is no smaller positive number k that has this property (where the counts are positive), so a(4)=8377774397163159586. There are 14 numbers with this property, the last one being 8377774397163162544. - _Henri Lifchitz_, Jan 31 2025

Cf. A005117, A072047.
Cf. 1 to 5 distinct primes: A000040, A006881, A007304, A046386, A046387.
Cf. 6 to 10 distinct primes: A067885, A123321, A123322, A115343, A281222.
Cf. A340316.

a(n) = my(nbm = 0, nbn = 0); for (k=1, oo, if (issquarefree(k), my(o=omega(k)); if (o==n, nbn++); if (o==n-1, nbm++); if (nbm && (nbn==nbm), return(k)))); \\ Michel Marcus, Nov 25 2022

from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A356683(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(k,n): return sum(primepi(k//prod(c[1] for c in a))-a[-1][0] for a in g(k,0,1,1,n)) if n>1 else primepi(k)
    return 2 if n==1 else next(k for k in count(1) if (x:=f(k,n-1))>0 and x==f(k,n)) # Chai Wah Wu, Aug 31 2024

a(4) from Henri Lifchitz, Jan 31 2025

A375418 Products of prime 6-tuples (p, p+4, p+6, p+10, p+12, p+16) where p = A022008(n).

Original entry on oeis.org

7436429, 1329900201629, 17190330954910965900632429, 53723718911110731187434029, 7046153584492008675489230429, 1688812201738097614580773379554136429, 17799106117345926490096695600218208629, 55722944657811823198723449024051143429, 505827208840254150110614056219371285429

Offset: 1

Michael De Vlieger, Aug 16 2024

nonn

Subsequence of A067885.
All terms are congruent to -1 (mod 30), since they are a product of the following residues (mod 30): {7, 11, 13, 17, 19, 23}.
All terms but the first are congruent to -1 (mod 210), since they are a product of the following residues (mod 210): {97, 101, 103, 107, 109, 113}; a(1) mod 210 = 119.

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

Cf. A022008, A067885.

Map[Times @@ NextPrime[#, Range[0, 5]] &, Select[Prime@ Range[2^20], AllTrue[{# + 4, # + 6, # + 10, # + 12, # + 16}, PrimeQ] &]]

cp's OEIS Frontend

A340467 a(n) is the n-th squarefree number having n prime factors.

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A346014 Numbers whose average number of distinct prime factors of their divisors is an integer.

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A214195 Numbers with the number of distinct prime factors a multiple of 3.

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A378097 Products of 6 distinct primes that are sandwiched between twin prime numbers.

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A378627 Products of 6 distinct primes that are sandwiched between semiprime numbers.

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A172443 Numbers with exactly 64 divisors.

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A282141 a(n)=least number strictly greater than n with an equivalent prime tower factorization.

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PARI

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A327829 Squarefree numbers with a prime number of prime factors.

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