A067885 -id:A067885 - cp's OEIS frontend

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

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

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

A176655 Numbers that are divisible by exactly 7 distinct primes.

Original entry on oeis.org

510510, 570570, 690690, 746130, 870870, 881790, 903210, 930930, 1009470, 1021020, 1067430, 1111110, 1138830, 1141140, 1193010, 1217370, 1231230, 1272810, 1291290, 1345890, 1360590, 1381380, 1385670, 1411410, 1438710

Offset: 1

Vladimir Joseph Stephan Orlovsky, Apr 22 2010

nonn

			1711710 = 2 * 3^2 * 5 * 7 * 11 * 13 * 19.

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

Row 7 of A125666.

Cf. A046386, A046387, A067885, A123321.

Select[Range[9!,5*9! ],Length[FactorInteger[ # ]]==7&]
Select[Range[144*10^4],PrimeNu[#]==7&] (* Harvey P. Dale, Jul 05 2022 *)

isA176655(n)=omega(n)==7 \\ Charles R Greathouse IV, Mar 11 2011

(PARI) A246655(lim)=my(v=List(primes([2,lim\=1]))); for(e=2,logint(lim,2), forprime(p=2,sqrtnint(lim,e), listput(v,p^e))); Set(v)
list(lim,pr=7)=if(pr==1, return(A246655(lim))); my(v=List(),pr1=pr-1,mx=prod(i=1,pr1,prime(i))); forprime(p=prime(pr),lim\mx, my(u=list(lim\p,pr1)); for(i=1,#u,listput(v,p*u[i]))); Set(v) \\ Charles R Greathouse IV, Feb 03 2023

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

Original entry on oeis.org

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

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.

A147573 Numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13}.

Original entry on oeis.org

30030, 60060, 90090, 120120, 150150, 180180, 210210, 240240, 270270, 300300, 330330, 360360, 390390, 420420, 450450, 480480, 540540, 600600, 630630, 660660, 720720, 750750, 780780, 810810, 840840, 900900, 960960, 990990, 1051050, 1081080, 1171170, 1201200, 1261260

Offset: 1

Artur Jasinski, Nov 07 2008

nonn

Successive numbers k such that EulerPhi(x)/x = m:
( Family of sequences for successive n primes )
m=1/2 numbers with exactly 1 distinct prime divisor {2} see A000079
m=1/3 numbers with exactly 2 distinct prime divisors {2,3} see A033845
m=4/15 numbers with exactly 3 distinct prime divisors {2,3,5} see A143207
m=8/35 numbers with exactly 4 distinct prime divisors {2,3,5,7} see A147571
m=16/77 numbers with exactly 5 distinct prime divisors {2,3,5,7,11} see A147572
m=192/1001 numbers with exactly 6 distinct prime divisors {2,3,5,7,11,13} see A147573
m=3072/17017 numbers with exactly 7 distinct prime divisors {2,3,5,7,11,13,17} see A147574
m=55296/323323 numbers with exactly 8 distinct prime divisors {2,3,5,7,11,13,17,19} see A147575
Although 39270 has exactly 6 distinct prime divisors (39270=2*3*5*7*11*17), it is not in this sequence because the 6 distinct prime divisors may only comprise 2, 3, 5, 7, 11, and 13. - Harvey P. Dale, Oct 11 2014

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

Subsequence of A067885 and of A080197.

Cf. A060735, A143207, A147571-A147575, A147576-A147580.

a = {}; Do[If[EulerPhi[x]/x == 192/1001, AppendTo[a, x]], {x, 1, 100000}]; a

is(n)=if(n%30030, return(0)); my(g=30030); while(g>1, n/=g; g=gcd(n,30030)); n==1 \\ Charles R Greathouse IV, Sep 14 2015

a(n) = 30030 * A080197(n). - Charles R Greathouse IV, Sep 14 2015

Sum_{n>=1} 1/a(n) = 1/5760. - Amiram Eldar, Nov 12 2020

More terms from Amiram Eldar, Mar 10 2020

A046396 Palindromes which are the product of 6 distinct primes.

Original entry on oeis.org

222222, 282282, 474474, 555555, 606606, 646646, 969969, 2040402, 2065602, 2206022, 2417142, 2646462, 2673762, 2875782, 3262623, 3309033, 4179714, 4192914, 4356534, 4585854, 4912194, 5021205, 5169615, 5174715, 5578755

Offset: 1

Patrick De Geest, Jun 15 1998

The original definition "Palindromes with exactly 6 distinct prime factors" was misleading. For example, the number 414414 = 2 * 3^2 * 7 * 11 * 13 * 23 has exactly 6 distinct prime factors, although the factor 3 occurs twice. But the listed terms show that it is not in this sequence. See sequence A373466 for the variant corresponding to that definition. - M. F. Hasler, Jun 06 2024

Cf. A046332 (similar, but for 6 prime factors counted with multiplicity).
Cf. A002113 (palindromes), A067885 (products of 6 distinct primes).
Cf. A074969 (numbers having 6 distinct prime divisors).

Select[Range[6*10^6],#==IntegerReverse[#]&&PrimeNu[#]==PrimeOmega[#]==6&] (* The program uses the IntegerReverse function from Mathematica version 10 *) (* Harvey P. Dale, Mar 17 2016 *)

A046332_upto(N, start=1, num_fact=6)={ my(L=List()); while(N >= start = nxt_A002113(start), omega(start)==num_fact && issquarefree(start) && listput(L, start)); L} \\ M. F. Hasler, Jun 06 2024

Intersection of A002113 and A067885. - M. F. Hasler, Jun 06 2024

Name edited

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.

A294776 Squarefree products of k primes that are symmetrically distributed around their average. Case k = 6.

Original entry on oeis.org

1616615, 3411705, 7436429, 9408035, 10163195, 12838371, 13037385, 13844919, 14969435, 19605131, 20414121, 23783045, 24997749, 25113935, 27568145, 30478565, 31473255, 32518535, 33999455, 39946569, 43134015, 46115135, 48215255, 50907855, 56179409, 61558343

Offset: 1

Paolo P. Lava, Nov 09 2017

nonn

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

Subsequence of A067885.

Cf. A006881 (k=2), A262723 (k=3), A294751 (k=4), A294752 (k=5).

with(numtheory): P:=proc(q,h) local a,b,k,n,ok;
for n from 2*3*5*7*11*13 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(10^9,6);
# Alternative:
N:= 10^8: # to get all terms <= N
M:= floor(fsolve(3*5*7*(M-7)*(M-5)*(M-3) = N)):
P:= select(isprime, [seq(i,i=3..M/2,2)]): nP:= nops(P):
Res:= NULL:
for m from 10 by 2 to M do
  for ix from 1 to nP-2 do
    x:= P[ix];
    if x >= m/2 or (x*(m-x))^3 >= N then break fi;
    if not isprime(m-x) then next fi;
    for iy from ix+1 to nP-1 do
      y:= P[iy];
      if y >= m/2 or x*(m-x)*(y*(m-y))^2 >= N then break fi;
      if not isprime(m-y) then next fi;
      for iz from iy+1 to nP do
        z:= P[iz];
        if z >= m/2 then break fi;
        v:= x*(m-x)*y*(m-y)*z*(m-z);
        if v > N then break fi;
        if isprime(m-z) then Res:= Res, v fi;
od od od od:
sort([Res]); # Robert Israel, May 19 2019

isok(n, nb=6) = {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

A168352 Products of 6 distinct odd primes.

Original entry on oeis.org

255255, 285285, 345345, 373065, 435435, 440895, 451605, 465465, 504735, 533715, 555555, 569415, 596505, 608685, 615615, 636405, 645645, 672945, 680295, 692835, 705705, 719355, 726495, 752115, 770385, 780045, 795795, 803985, 805035, 811965, 823515, 838695, 844305, 858585

Offset: 1

Vladimir Joseph Stephan Orlovsky, Nov 23 2009

			255255 = 3*5*7*11*13*17
285285 = 3*5*7*11*13*19
345345 = 3*5*7*11*13*23
435435 = 3*5*7*11*13*29

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

Cf. A005408, A067885.

Cf. A046391 (5 distinct odd primes).

f[n_]:=Last/@FactorInteger[n]=={1,1,1,1,1,1}&&FactorInteger[n][[1,1]]>2; lst={};Do[If[f[n],AppendTo[lst,n]],{n,6*9!}];lst

is(n) = {n%2 == 1 && factor(n)[,2]~ == [1,1,1,1,1,1]} \\ David A. Corneth, Aug 26 2020

from sympy import primefactors, factorint
print([n for n in range(1, 1000000, 2) if len(primefactors(n)) == 6 and max(list(factorint(n).values())) == 1]) # Karl-Heinz Hofmann, Mar 01 2023

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A168352(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,1,2,1,6)))
    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,n,n) # Chai Wah Wu, Sep 10 2024

A067885 INTERSECT A005408. [R. J. Mathar, Nov 24 2009]

Definition corrected by R. J. Mathar, Nov 24 2009

More terms from David A. Corneth, Aug 26 2020

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