A046387 -id:A046387 - cp's OEIS frontend

A123322 Products of 8 distinct primes (squarefree 8-almost primes).

9699690, 11741730, 13123110, 14804790, 15825810, 16546530, 17160990, 17687670, 18888870, 20030010, 20281170, 20930910, 21111090, 21411390, 21637770, 21951930, 23130030, 23393370, 23993970, 24534510, 25555530, 25571910

Offset: 1

Rick L. Shepherd, Sep 25 2006

nonn

Intersection of A005117 and A046310.

			a(1) = 9699690 = 2*3*5*7*11*13*17*19 = A002110(8).

Robert G. Wilson v, Table of n, a(n) for n = 1..29422 (first 10000 terms from Rick Shepherd)
Index to sequences related to prime signature

Cf. A001221, A001222, A005117, A046310, A048692, Squarefree k-almost primes: A000040 (k=1), A006881 (k=2), A007304 (k=3), A046386 (k=4), A046387 (k=5), A067885 (k=6), A123321 (k=7), A115343 (k=9).

N:= 3*10^7: # to get all terms  <= N
pmax:= floor(N/mul(ithprime(i),i=1..7)):
Primes:= select(isprime,[2,seq(i,i=3..pmax,2)]):
sort(select(`<`,map(convert,combinat:-choose(Primes,8),`*`),N)); # Robert Israel, Dec 18 2018

f8Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1, 1, 1, 1}; lst={};Do[If[f8Q[n], AppendTo[lst, n]], {n, 10!, 11!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)
Take[ Sort[ Times @@@ Subsets[ Prime@ Range@ 15, {8}]], 22] (* Robert G. Wilson v, Dec 18 2018 *)

is(n)=issquarefree(n)&&omega(n)==8 \\ Charles R Greathouse IV, Feb 01 2017, corrected (following an observation from Zak Seidov) by M. F. Hasler, Dec 19 2018

is(n) = my(f = factor(n)); omega(f) == 8 && bigomega(f) == 8 \\ David A. Corneth, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A123322(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,0,1,1,8)))
    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

Edited by Robert Israel, Dec 18 2018

A259349 Numbers n such that n-1, n, and n+1 are all products of 6 distinct primes (i.e. belong to A067885).

Original entry on oeis.org

1990586014, 1994837494, 2129658986, 2341714794, 2428906514, 2963553594, 3297066410, 3353808094, 3373085990, 3623442746, 3659230730, 3809238770, 3967387346, 4058711734, 4144727994, 4196154390, 4502893746, 4555267690, 4653623534

Offset: 1

James G. Merickel, Jun 24 2015

nonn

A subsequence of A169834 and A067885.
The rudimentary method employed by the PARI program below reaches the limit of its usefulness here. Contrast it with the method required for A259350, which is over 4.5 orders of magnitude faster than the analog of this (and may still be some distance best).
a(1)=A093550(6) (that sequence's 5th term, with offset 2). The program arbitrarily makes use of this knowledge, but will run (slower) without it.

			1990586013 = 3*13*29*67*109*241,
1990586014 = 2*23*37*43*59*461, and
1990586015 = 5*11*17*19*89*1259; and no smaller trio of this kind exists, making the middle value a(1).

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A093550, A169834, A364265, A248201, A248202, A248203, A248204, A259350, A259801.
For products of 1, 2, 3, 4, 5, and 6 distinct primes see A000040, A006881, A007304, A046386, A046387, and A067885, resp.
See A364265 for a closely related sequence. - N. J. A. Sloane, Jul 18 2023

{
\\Program initialized with known a(1).\\
\\The purpose of vector s and value u\\
\\is to skip bad values modulo 36.\\
k=1990586014;s=[4,4,8,8,8,4];u=1;
while(1,
  if(issquarefree(k),
    if(issquarefree(k-1),
      if(issquarefree(k+1),
        if(omega(k)==6,
          if(omega(k-1)==6,
            if(omega(k+1)==6,
              print1(k" ")))))));
  k+=s[u];if(u==6,u=1,u++))
}

{n: A001221(n-1) = A001221(n) = A001221(n+1) = A001222(n-1) = A001222(n) = A001222(n+1) = 6}. - R. J. Mathar, Jul 18 2023

A192203 Numbers k such that k, k+1, and k+2 are each the product of exactly 5 distinct primes.

Original entry on oeis.org

16467033, 18185869, 21134553, 21374353, 21871365, 22247553, 22412533, 22721585, 24845313, 25118093, 25228929, 25345333, 25596933, 26217245, 27140113, 29218629, 29752345, 30323733, 30563245, 31943065, 32663265, 33367893, 36055045, 38269021, 39738061, 40547065

Offset: 1

Gil Broussard, Jun 25 2011

nonn

Numbers k such that k, k+1, and k+2 are all members of A046387. - N. J. A. Sloane, Jul 17 2024
A subsequence of A242608 intersect A016813. - M. F. Hasler, May 19 2014
All terms are congruent to 1 mod 4. - Zak Seidov, Dec 22 2014

			a(1)=16467033 because it is the product of 5 distinct primes (3,11,17,149,197), and so are a(1)+1: 16467034 (2,19,23,83,227), and a(1)+2: 16467035 (5,13,37,41,167).

Zak Seidov, Table of n, a(n) for n = 1..154

Cf. A046387, A140079. Subsequence of A318964 and of A364266.

Cf. A039833, A066509, A176167.

SequencePosition[Table[If[PrimeNu[n]==PrimeOmega[n]==5,1,0],{n,164*10^5,406*10^5}],{1,1,1}][[;;,1]]+164*10^5-1 (* Harvey P. Dale, Jul 17 2024 *)

forstep(n=1+10^7,1e8,4, for(k=n,n+2,issquarefree(k)||next(2)); for(k=n,n+2,omega(k)==5||next(2));print1((n)", ")) \\ M. F. Hasler, May 19 2014

A340316 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where row n is the increasing list of all squarefree numbers with n primes.

Original entry on oeis.org

2, 3, 6, 5, 10, 30, 7, 14, 42, 210, 11, 15, 66, 330, 2310, 13, 21, 70, 390, 2730, 30030, 17, 22, 78, 462, 3570, 39270, 510510, 19, 26, 102, 510, 3990, 43890, 570570, 9699690, 23, 33, 105, 546, 4290, 46410, 690690, 11741730, 223092870

Offset: 1

Peter Dolland, Jan 04 2021

This is a permutation of all squarefree numbers > 1.

			First six rows and columns:
      2     3     5     7    11    13
      6    10    14    15    21    22
     30    42    66    70    78   102
    210   330   390   462   510   546
   2310  2730  3570  3990  4290  4830
  30030 39270 43890 46410 51870 53130

Cf. A005117 (squarefree numbers), A072047 (number of prime factors), A340313 (indexing), A078840 (all natural numbers, not only squarefree).
Rows n=1..10: A000040, A006881, A007304, A046386, A046387, A067885, A123321, A123322, A115343, A281222.
Columns k=1..2: A002110, A306237.
Main diagonal gives A340467.
Cf. A358677.

a340316 n k = a340316_row n !! (k-1)
a340316_row n = [a005117_list !! k | k <- [0..], a072047_list !! k == n]

from math import prod, isqrt
from sympy import prime, primerange, integer_nthroot, primepi
def A340316_T(n,k):
    if n == 1: return prime(k)
    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(k+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(A072047(n), A340313(n)) = A005117(n) for n > 1.

A087978 a(n) is the first term in a chain of at least n consecutive numbers, each having exactly m = 5 distinct prime factors.

Original entry on oeis.org

2310, 254540, 1042404, 21871365, 129963314, 830692265, 4617927894, 18297409143, 41268813542, 287980277114, 1182325618032, 6455097761454, 14207465691240, 54049709480208, 90987640183352, 546525829796442, 546525829796442

Offset: 1

Labos Elemer, Sep 26 2003

Every chain of 30030 consecutive numbers has exactly one number divisible by 30030 = 2 * 3 * 5 * 7 * 11 * 13 hence is divisible by more than five distinct primes. Therefore the sequence is finite. - David A. Corneth, Jul 19 2023

a(18) > 2 * 10^15. - Toshitaka Suzuki, Jun 23 2025

Roger B. Eggleton and James A. MacDougall, Consecutive Integers with Equally Many Principal Divisors, Math. Mag. 81 (2008), 235-248. [_T. D. Noe_, Oct 13 2008]

Cf. A064708 (m=2), A080569 (m=3), A087977 (m=4).
Cf. A138206, A138207, A154573. - Donovan Johnson, Jan 15 2009
Cf. A046387.

k=1; Do[While[Union[Table[Length[FactorInteger[i]], {i, k, k+n-1}]]!={5}, k++ ]; Print[k], {n, 1, 8}]

More terms from Don Reble, Sep 29 2003
a(7)-a(10) from Donovan Johnson, Mar 06 2008
a(11)-a(12) from Donovan Johnson, Jan 15 2009
a(13)-a(15) from Toshitaka Suzuki, Apr 06 2025
a(16)-a(17) from Toshitaka Suzuki, Jun 23 2025

A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

Original entry on oeis.org

378014, 421134, 483405, 486590, 486794, 489345, 507129, 545258, 549185, 558789, 558830, 634809, 637329, 663585, 667029, 690234, 720290, 776985, 782690, 823745, 824109, 853005, 853034, 855645, 873885, 883245, 892905, 935714, 945230, 968253, 987734, 999005, 1005081, 1013726

Offset: 1

Seiichi Manyama, Sep 06 2018

nonn

			n | a(n)                           | a(n)+1
--+--------------------------------+--------------------------------
1 | 378014 = 2 * 7 * 13 * 31 *  67 | 378015 = 3 *  5 * 11 * 29 * 79
2 | 421134 = 2 * 3 *  7 * 37 * 271 | 421135 = 5 * 11 * 13 * 19 * 31
3 | 483405 = 3 * 5 * 13 * 37 *  67 | 483406 = 2 *  7 * 11 * 43 * 73

Seiichi Manyama, Table of n, a(n) for n = 1..1000

Subsequence of A140079.

Cf. A046387, A052215, A215217, A263990, A318896.

is(n) = omega(n)==5 && omega(n+1)==5 && bigomega(n)==5 && bigomega(n+1)==5 \\ Felix Fröhlich, Sep 06 2018

A046395 Palindromes that are the product of 5 distinct primes.

Original entry on oeis.org

6006, 8778, 20202, 28182, 41514, 43134, 50505, 68586, 87978, 111111, 141141, 168861, 202202, 204402, 209902, 246642, 249942, 262262, 266662, 303303, 323323, 393393, 399993, 438834, 454454, 505505, 507705, 515515, 516615, 519915, 534435, 535535, 543345

Offset: 1

Patrick De Geest, Jun 15 1998

No exponent of the distinct prime factors can be greater than one, i.e., no prime powers are permitted. - Harvey P. Dale, Apr 09 2021 at the suggestion of Sean A. Irvine

See A373465 for the similar sequence where only distinct prime divisors are counted, but may occur to higher powers. - M. F. Hasler, Jun 06 2024

			505505 = 5 * 7 * 11 * 13 * 101.

Harvey P. Dale, Table of n, a(n) for n = 1..500

Cf. A002113 (palindromes), A051270 (omega(.) = 5).

Cf. A046331 (palindromes with 5 prime factors counted with multiplicity), A373465 (counting only distinct prime divisors).

Select[Range[550000],PalindromeQ[#]&&PrimeNu[#]==PrimeOmega[#]==5&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 09 2021 *)

Intersection of A002113 and A046387.

Corrected at the suggestion of Sean A. Irvine by Harvey P. Dale, Apr 09 2021

Name edited to avoid confusion by M. F. Hasler, Jun 06 2024

A279686 Numbers that are the least integer of a prime tower factorization equivalence class (see Comments for details).

Original entry on oeis.org

1, 2, 4, 6, 8, 12, 16, 18, 30, 36, 40, 48, 60, 64, 72, 81, 90, 108, 144, 162, 180, 192, 200, 210, 225, 240, 256, 280, 320, 324, 360, 405, 420, 432, 450, 500, 512, 540, 576, 600, 630, 648, 720, 768, 810, 900, 960, 1260, 1280, 1296, 1350, 1400, 1536, 1575, 1600

Offset: 1

Rémy Sigrist, Dec 16 2016

nonn

The prime tower factorization of a number is defined in A182318.
We say that two numbers, say n and m, belong to the same prime tower factorization equivalence class iff there is a permutation of the prime numbers, say f, such that replacing each prime p by f(p) in the prime tower factorization of n leads to m.
The notion of prime tower factorization equivalence class can be seen as a generalization of the notion of prime signature; thereby, this sequence can be seen as an equivalent of A025487.
This sequence contains all primorial numbers (A002110).
This sequence contains A260548.
This sequence contains the terms > 0 in A014221.
If n appears in the sequence, then 2^n appears in the sequence.
If n appears in the sequence and k>=0, then A002110(k)^n appears in the sequence.
With the exception of term 1, this sequence contains no term from A182318.
Odd numbers appearing in this sequence: 1, 81, 225, 405, 1575, 2025, 2835, 6125, 10125, 11025, 14175, 15625, 16875, 17325, 31185, 33075, 50625, 67375, 70875, 99225, ...
Here are some prime tower factorization equivalence classes:
- Class 1: the number one (the only finite equivalence class),
- Class p: the prime numbers (A000040),
- Class p*q: the squarefree semiprimes (A006881),
- Class p^p: the numbers of the form p^p with p prime (A051674),
- Class p^q: the numbers of the form p^q with p and q distinct primes,
- Class p*q*r: the sphenic numbers (A007304),
- Class p*q*r*s: the products of four distinct primes (A046386),
- Class p*q*r*s*t: the products of five distinct primes (A046387),
- Class p*q*r*s*t*u: the products of six distinct primes (A067885).

			2 is the least number of the form p with p prime, hence 2 appears in the sequence.
6 is the least number of the form p*q with p and q distinct primes, hence 6 appears in the sequence.
72 is the least number of the form p^q*q^p with p and q distinct primes, hence 72 appears in the sequence.
36000 is the least number of the form p^q*q^r*r^p with p, q and r distinct primes, hence 36000 appears in the sequence.

Rémy Sigrist, Table of n, a(n) for n = 1..1000
Roberto Conti and Pierluigi Contucci, A Natural Avenue, arXiv:2204.08982 [math.NT], 2022.
Rémy Sigrist, PARI program for A279686
Rémy Sigrist, Prime tower factorization of the first terms

Cf. A000040, A002110, A006881, A007304, A014221, A025487, A046386, A046387, A051674, A067885, A182318, A260548, A279690.

A294752 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 5.

Original entry on oeis.org

53295, 119301, 229245, 399993, 608235, 623645, 1462731, 2324495, 3696189, 3973145, 4482879, 5356445, 5920971, 6249633, 7588977, 8270385, 10160943, 10450121, 10505373, 13185969, 13630011, 13760929, 14935029, 19095395, 20280795, 22566271, 23131549, 23408259, 24778401

Offset: 1

Paolo P. Lava, Nov 08 2017

nonn

			53295 = 3*5*11*17*19. Prime factors average is (3 + 5 + 11 + 17 + 19)/5 = 11 and 3 + 8 = 11 = 19 - 8, 5 + 6 = 11 = 17 - 6.

Robert Israel, Table of n, a(n) for n = 1..3560

Subsequence of A046387, A203614.

Cf. A006881 (k=2), A262723 (k=3), A294751 (k=4), A294776 (k=6).

with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7*11 to q do if not isprime(n) and issqrfree(n) then a:=ifactors(n)[2];
if nops(a)=h then b:=2*add(a[k][1],k=1..nops(a))/nops(a); ok:=1;
for k from 1 to trunc(nops(a)/2) do if a[k][1]+a[nops(a)-k+1][1]<>b then ok:=0; break; fi; od; if ok=1 then print(n); fi; fi; fi; od; end: P(10^9,5);
# Alternative:
N:= 10^8: # to get all terms <= N
M:= floor((8*N/15)^(1/3)):
P:= select(isprime, [seq(i,i=3..M,2)]): nP:= nops(P):
Res:= NULL:
for i3 from 3 to nP-2 do
  p3:= P[i3];
  for i1 from 1 to i3-2 do
    if isprime(2*p3 - P[i1]) then
      for i2 from i1+1 to i3-1 do
        if isprime(2*p3 - P[i2]) then
          v:=P[i1]*P[i2]*p3*(2*p3-P[i2])*(2*p3-P[i1]);
          if v <= N then Res:= Res, v fi
        fi
      od
     fi
   od
od:
sort([Res]): # Robert Israel, Nov 10 2017

isok(n, nb=5) = {if (issquarefree(n) && (omega(n)==nb), f = factor(n)[, 1]~; avg = vecsum(f)/#f; for (k=1, #f\2, if (f[k] + f[#f-k+1] != 2*avg, return(0));); return (1););} \\ Michel Marcus, Nov 10 2017

More terms from Giovanni Resta, Nov 09 2017

Missing term 23131549 inserted by Robert Israel, Nov 10 2017

A376380 Products of 5 distinct primes that are sandwiched between twin prime numbers.

Original entry on oeis.org

2310, 2730, 6090, 6270, 7590, 8970, 9282, 13398, 14322, 15330, 17490, 19470, 21318, 22110, 23370, 27690, 28182, 29670, 30090, 32190, 32370, 32718, 32802, 32970, 33330, 37590, 40530, 41610, 45318, 46830, 47058, 48678, 48990, 49170, 49530, 49938, 51198, 52710, 56238, 56910, 57270, 58110, 58170, 59010, 60762

Offset: 1

Massimo Kofler, Sep 22 2024

nonn

All terms are even.

All terms are of the form 6r, where r is coprime to 6, so they all are Zumkeller numbers (A083207). - Ivan N. Ianakiev, Sep 24 2024

			2310 is in the sequence a term because 2310=2*3*5*7*11 is the product of five distinct primes and 2309, 2311 are a couple of twin primes.
2730 is in the sequence a term because 2730=2*3*5*7*13 is the product of five distinct primes and 2729, 2731 are a couple of twin primes.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Intersection of A014574 and A046387.

Cf. A353022.

ispenta:= proc(n) local F;
  F:= ifactors(n)[2];
  nops(F) = 5 and F[..,2] = [1$5]
end proc:
select(t -> isprime(t-1) and isprime(t+1) and ispenta(t), [seq(i,i=6 .. 10^5,12)]); # Robert Israel, Sep 24 2024

Select[Range[6, 61000, 6], And @@ PrimeQ[# + {-1, 1}] && FactorInteger[#][[;; , 2]] == {1, 1, 1, 1, 1} &] (* Amiram Eldar, Sep 22 2024 *)

list(lim)=my(v=List()); forprime(p=11,lim\210, my(P=lim\(6*p)); forprime(q=7,min(P\5,p-2), my(Q=P\q); forprime(r=5,min(Q,q-2), my(t=6*p*q*r); if(isprime(t+1) && isprime(t-1), listput(v,t))))); Set(v) \\ Charles R Greathouse IV, Sep 24 2024

cp's OEIS Frontend

A123322 Products of 8 distinct primes (squarefree 8-almost primes).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

PARI

Python

Extensions

A259349 Numbers n such that n-1, n, and n+1 are all products of 6 distinct primes (i.e. belong to A067885).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

A192203 Numbers k such that k, k+1, and k+2 are each the product of exactly 5 distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

A340316 Square array A(n,k), n>=1, k>=1, read by antidiagonals, where row n is the increasing list of all squarefree numbers with n primes.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Haskell

Python

Formula

A087978 a(n) is the first term in a chain of at least n consecutive numbers, each having exactly m = 5 distinct prime factors.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Extensions

A318964 Numbers k such that both k and k+1 are the product of exactly five distinct primes.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

A046395 Palindromes that are the product of 5 distinct primes.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

Extensions