A046386 -id:A046386 - cp's OEIS frontend

A067885 Products of exactly 6 distinct primes.

30030, 39270, 43890, 46410, 51870, 53130, 62790, 66990, 67830, 71610, 72930, 79170, 81510, 82110, 84630, 85470, 91770, 94710, 98670, 99330, 101010, 102102, 103530, 106590, 108570, 110670, 111930, 114114, 115710, 117390, 122430, 123690, 124410, 125970, 128310

Offset: 1

Benoit Cloitre, Mar 02 2002

nonn

Subsequence of A074969. - R. J. Mathar, Nov 24 2009

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.

Select[Range[125000],PrimeNu[#]==PrimeOmega[#]==6&] (* Harvey P. Dale, May 14 2014 *)

is(n)=factor(n)[,2]==[1,1,1,1,1,1]~ \\ Charles R Greathouse IV, Sep 14 2015

is(n)=omega(n)==6 && bigomega(n)==6 \\ Hugo Pfoertner, Dec 18 2018

list(lim)=lim\=1; my(v=List(), L1,L2,L3,L4,P4,P5); forprime(p=13,lim\2310, L1=lim\p; forprime(q=11,min(L1\210,p-2), L2=L1\q; forprime(r=7, min(L2\30,q-2), L3=L2\r; forprime(s=5,min(L3\6,r-2), L4=L3\s; P4=p*q*r*s; forprime(t=3, min(L4\2,s-2), P5=P4*t; forprime(u=2, min(L4\t,t-1), listput(v,P5*u))))))); Set(v) \\ Charles R Greathouse IV, Aug 27 2021

from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A067885(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,0,1,1,6)))
    kmin, kmax = 0,1
    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 # Chai Wah Wu, Aug 29 2024

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

A085987 Product of exactly four primes, three of which are distinct (p^2qr).

Original entry on oeis.org

60, 84, 90, 126, 132, 140, 150, 156, 198, 204, 220, 228, 234, 260, 276, 294, 306, 308, 315, 340, 342, 348, 350, 364, 372, 380, 414, 444, 460, 476, 490, 492, 495, 516, 522, 525, 532, 550, 558, 564, 572, 580, 585, 620, 636, 644, 650, 666, 693, 708, 726

Offset: 1

Alford Arnold, Jul 08 2003

nonn

A014613 is completely determined by A030514, A065036, A085986, A085987 and A046386 since p(4) = 5. (cf. A000041). More generally, the first term of sequences which completely determine the k-almost primes can be found in A036035 (a resorted version of A025487).

A050326(a(n)) = 4. - Reinhard Zumkeller, May 03 2013

			a(1) = 60 since 60 = 2*2*3*5 and has three distinct prime factors.

Cf. A001248, A006881, A030078, A054753, A007304, A050997, A046387, A036035, A086974.

Subsequence of A014613, A307341, A178212.

f[n_]:=Sort[Last/@FactorInteger[n]]=={1,1,2}; Select[Range[2000], f] (* Vladimir Joseph Stephan Orlovsky, May 03 2011 *)
pefp[{a_,b_,c_}]:={a^2 b c,a b^2 c,a b c^2}; Module[{upto=800},Select[ Flatten[ pefp/@Subsets[Prime[Range[PrimePi[upto/6]]],{3}]]//Union,#<= upto&]] (* Harvey P. Dale, Oct 02 2018 *)

list(lim)=my(v=List(),t,x,y,z);forprime(p=2,lim^(1/4),t=lim\p^2;forprime(q=p+1,sqrtint(t),forprime(r=q+1,t\q,x=p^2*q*r;y=p*q^2*r;listput(v,x);if(y<=lim,listput(v,y);z=p*q*r^2;if(z<=lim,listput(v,z))))));vecsort(Vec(v)) \\ Charles R Greathouse IV, Jul 15 2011

is(n)=vecsort(factor(n)[,2]~)==[1,1,2] \\ Charles R Greathouse IV, Oct 19 2015

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A085987(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((t:=primepi(s:=isqrt(y:=x//r**2)))+(t*(t-1)>>1)-sum(primepi(y//k) for k in primerange(1, s+1)) for r in primerange(isqrt(x)+1))+sum(primepi(x//p**3) for p in primerange(integer_nthroot(x,3)[0]+1))-primepi(integer_nthroot(x,4)[0])
    return bisection(f,n,n) # Chai Wah Wu, Mar 27 2025

More terms from Reinhard Zumkeller, Jul 25 2003

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

A074379 Carmichael numbers with exactly 4 prime factors.

Original entry on oeis.org

41041, 62745, 63973, 75361, 101101, 126217, 172081, 188461, 278545, 340561, 449065, 552721, 656601, 658801, 670033, 748657, 838201, 852841, 997633, 1033669, 1082809, 1569457, 1773289, 2100901, 2113921, 2433601, 2455921

Offset: 1

Jani Melik, Sep 24 2002

nonn

Original name was: "Super-Carmichael numbers with exactly 4 factors", and a comment explained that the prefix "super" means that the Moebius function (A008683) equals mu(N) = +1 for these. But for squarefree numbers such as Carmichael numbers (A002997), this just means that they have an even number of prime factors, which is trivial if that number is 4.

In the literature there are other definitions of "super-Carmichael numbers", see the McIntosh and Meštrović references, so we prefer not to use this terminology at all.

			41041 = 7 * 11 * 13 * 41.
62745 = 3 * 5 * 47 * 89.

R. J. Mathar and Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..6042 from R. J. Mathar)
Richard J. McIntosh Carmichael numbers with (p + 1) | (n - 1), Integers 14 (2014) #A59.
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867, May 04 2013

Cf. A002997 (Carmichael numbers), A006931 (least Carmichael with n prime factors), A046386 (products of four distinct primes).

p = Table[ Prime[i], {i, 1, 10}]; f[n_] := Union[ PowerMod[ Select[p, GCD[ #, n] == 1 & ], n - 1, n]]; Select[ Range[2500000], !PrimeQ[ # ] && OddQ[ # ] && Length[ FactorInteger[ # ]] == 4 && MoebiusMu[ # ] == 1 && f[ # ] == {1} & ]

is_A074379(n)=is_A002997(n) && is_A046386(n) \\ M. F. Hasler, Mar 24 2022

list(lim)=my(v=List()); forprime(p=3,sqrtnint(lim\=1,4), forprime(q=p+2,sqrtnint(lim\p,3), if(q%p==1, next); forprime(r=q+2,sqrtint(lim\p\q), if(r%p==1 || r%q==1, next); my(m=lcm([p-1,q-1,r-1]),pqr=p*q*r,t=Mod(1,m)/pqr,L=lim\pqr); fordiv(pqr-1,d, my(s=d+1); if(s>L, break); if(s==t && s>r && isprime(s), listput(v,pqr*s)))))); Set(v) \\ Charles R Greathouse IV, Apr 23 2022

Intersection of A002997 (Carmichael numbers) and A046386 (product of four distinct primes). - M. F. Hasler, Mar 24 2022

Edited and extended by Robert G. Wilson v, Oct 03 2002

Edited by M. F. Hasler, Mar 24 2022

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

A238748 Numbers k such that each integer that appears in the prime signature of k appears an even number of times.

Original entry on oeis.org

1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 36, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95, 100, 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

Offset: 1

Matthew Vandermast, May 08 2014

nonn

Values of n for which all numbers in row A238747(n) are even. Also, numbers n such that A000005(n^m) is a perfect square for all nonnegative integers m; numbers n such that A181819(n) is a perfect square; numbers n such that A182860(n) is odd.

The numbers of terms not exceeding 10^k, for k = 1, 2, ..., are 3, 33, 314, 3119, 31436, 315888, 3162042, 31626518, 316284320, 3162915907, ... . Apparently, the asymptotic density of this sequence exists and equals 0.3162... . - Amiram Eldar, Nov 28 2023

			The prime signature of 36 = 2^2 * 3^2 is {2,2}. One distinct integer (namely, 2) appears in the prime signature, and it appears an even number of times (2 times). Hence, 36 appears in the sequence.
The prime factorization of 1260 = 2^2 * 3^2 * 5^1 * 7^1. Exponent 2 occurs twice (an even number of times), as well as exponent 1, thus 1260 is included. It is also the first term k > 1 in this sequence for which A182850(k) = 4, not 3. - _Antti Karttunen_, Feb 06 2016

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

Subsequences include A006881, A030229, A046386, A077448, A162142, A179690, A190108, A190465, A367590.
Subsequence of A000379, A028260, A063774 and A268390.
Cf. A000005, A181819, A182850, A182860, A238747.

q[n_] := n == 1 || AllTrue[Tally[FactorInteger[n][[;; , 2]]][[;; , 2]], EvenQ]; Select[Range[200], q] (* Amiram Eldar, Nov 28 2023 *)

is(n) = {my(e = factor(n)[, 2], m = #e); if(m%2, return(0)); e = vecsort(e); forstep(i = 1, m, 2, if(e[i] != e[i+1], return(0))); 1;} \\ Amiram Eldar, Nov 28 2023

(define A238748 (MATCHING-POS 1 1 (lambda (n) (square? (A181819 n)))))
(define (square? n) (not (zero? (A010052 n))))
;; Requires also MATCHING-POS macro from my IntSeq-library - Antti Karttunen, Feb 06 2016

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

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.

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