A000977 -id:A000977 - cp's OEIS frontend

A264780 a(n) is the index of A246947(n) in A000977.

1, 3, 8, 13, 19, 26, 33, 2, 7, 14, 23, 43, 57, 67, 81, 93, 5, 18, 52, 72, 115, 138, 164, 10, 63, 127, 200, 240, 41, 49, 58, 66, 4, 16, 31, 47, 85, 107, 20, 60, 159, 214, 11, 34, 100, 134, 175, 22, 118, 234, 296, 75, 84, 6, 21, 39, 62, 109, 133, 27, 76, 197, 265

Offset: 1

Michel Marcus, Nov 24 2015

nonn

Is this a permutation of the positive integers?

			The first 5 terms of A246947 are 30, 60, 90, 120, and 150, that is, the 1st, 3rd, 8th, 13th and 19th terms of A000977.

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

Cf. A000977, A246947, A264762.

N:= 1000: # get all terms before the first term > N
W:= Vector(N, t -> if nops(numtheory:-factorset(t))<=2 then 0 else 1 fi):
WS:= ListTools:-PartialSums(convert(W, list)):
m:= 1:
F:= {2, 3,5}:
A[1]:= WS[30]:
W[30]:= 0:
for n from 2 do
    while W[m] = 0 and m < N do m:= m+1 od;
  for k from m to N do
     if W[k] = 1 and nops(numtheory:-factorset(k) intersect F) = 3 then
        A[n]:= WS[k];
        W[k]:= 0;
        F:= numtheory:-factorset(k);
        break
     fi
  od;
  if k > N then break fi;
od:
seq(A[i],i=1..n-1); # Robert Israel, Nov 26 2015

v246947(nn) = {a = 30; fa = (factor(a)[,1])~; va = [a]; vs = va; k = 0; while (k < nn, k = 1; while (!((#setintersect(fa, (factor(k)[,1])~) == 3) && (! vecsearch(vs, k))), k++); a = k; fa = (factor(a)[,1])~; va = concat(va, k); vs = vecsort(va);); va;}
v000977(nn) = {va = []; for (n=1, nn, if (omega(n) >= 3, va = concat(va, n));); va;}
lista(nn) = {v = v246947(nn); w = v000977(vecmax(v)); for (k=1, #v, for (j=1, #w, if (w[j] == v[k], print1(j, ", "); break);););}

A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 3, 2, 2, 2, 2, 3, 2, 2, 3, 2, 2, 2, 3, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 3, 2, 2, 3, 3, 3, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 3, 2, 5, 2, 3, 3, 2, 2, 2, 2, 2, 3, 2, 5, 3, 3, 3, 2, 2, 2, 2, 3, 2, 3, 2, 3, 2, 2, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 3, 3, 2, 2, 2, 2, 3, 5, 2, 2, 3, 2, 3

Offset: 1

Jonathan Vos Post, Jan 12 2011

Third largest prime factor of numbers k such that omega(k) = A001221(k) > 2. The 3rd largest prime factor may equal the second largest. This is not identical to third largest distinct prime factor of numbers that are divisible by at least three different primes. Indices n where a(n) equals 2, 3, 5, 7, 11, 13, 17, 19, 23, ... for the first time are 1, 8, 72, 299, 905, 1718, 3302, 6020, 10330, ... the corresponding numbers from A000977 are 30, 90, 350, 1001, 2431, 4199, 7429, 12673, 20677, ...

			a(1) = 2 because 30 = 2 * 3 * 5 has third largest prime factor 2.
a(2) = 2 because 42 = 2 * 3 * 7 has third largest prime factor 2.
a(3) = 2 because 60 = 2 * 2 * 3 * 5 has both third and fourth largest prime factor 2.
a(8) = 3 because 90 = 2 * 3 * 3 * 5 has both second and third largest prime factor 3.

Cf. A000040, A000977, A001221, A002808, A033992, A007774, A000961, A033993, A051270, A087040, A088739, A179312.

b:= proc(n) option remember; local k;
      if n=1 then 30
      else for k from b(n-1)+1 while
              nops(ifactors(k)[2])<3 do od;
           k
      fi
    end:
a:= n-> sort(map(x-> x[1]$x[2], ifactors(b(n))[2]))[-3]:
seq(a(n), n=1..120);

b[n_] := b[n] = Module[{k}, If[n==1, 30, For[k = b[n-1]+1, PrimeNu[k] < 3, k++]; k]];
a[n_] := (Table[#[[1]], {#[[2]]}]& /@ FactorInteger[b[n]] // Flatten // Sort)[[-3]];
Array[a, 120] (* Jean-François Alcover, Nov 28 2020, after Alois P. Heinz *)

Edited by Alois P. Heinz, Jan 14 2011

A046387 Products of exactly 5 distinct primes.

Original entry on oeis.org

2310, 2730, 3570, 3990, 4290, 4830, 5610, 6006, 6090, 6270, 6510, 6630, 7410, 7590, 7770, 7854, 8610, 8778, 8970, 9030, 9282, 9570, 9690, 9870, 10010, 10230, 10374, 10626, 11130, 11310, 11730, 12090, 12210, 12390, 12558, 12810, 13090, 13110

Offset: 1

Patrick De Geest, Jun 15 1998

Subsequence of A051270. 4620 = 2^2*3*5*7*11 is in A051270 but not in here, for example. - R. J. Mathar, Nov 10 2014

			a(1) = 2310 = 2 * 3 * 5 * 7 * 11 = A002110(5) = 5#.
a(2) = 2730 = 2 * 3 * 5 * 7 * 13.
a(3) = 3570 = 2 * 3 * 5 * 7 * 17.
a(10) = 6006 = 2 * 3 * 7 * 11 * 13.

T. D. Noe, Table of n, a(n) for n = 1..10000

Products of exactly k distinct primes, for k = 1 to 6: A000040, A006881. A007304, A046386, A046387, A067885.
Cf. A000040, A000961, A001221, A005117, A000977, A002110, A006881, A007304, A007774, A033992, A033993, A046386.
Cf. A014614, A046403, A051270.

A046387 := proc(n)
    option remember;
    local a;
    if n = 1 then
        2*3*5*7*11 ;
    else
        for a from procname(n-1)+1 do
            if A001221(a)= 5 and issqrfree(a) then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Oct 13 2019

f5Q[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1, 1}; lst={};Do[If[f5Q[n], AppendTo[lst, n]], {n, 8!}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 26 2008 *)

is(n)=factor(n)[,2]==[1,1,1,1,1]~ \\ Charles R Greathouse IV, Sep 17 2015

is(n)= omega(n)==5 && bigomega(n)==5 \\ Hugo Pfoertner, Dec 18 2018

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A046387(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,0,1,1,5)))
    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 30 2024

Entry revised by N. J. A. Sloane, Apr 10 2006

A033992 Numbers that are divisible by exactly three different primes.

Original entry on oeis.org

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, 286

Offset: 1

Labos Elemer

nonn

This sequence and A000977 are identical through their first 32 terms, but A000977(33) = 210. [Comment edited by Jon E. Schoenfield, Dec 30 2014]

			220 = 2*2*5*11 is here but 210 = 2*3*5*7 is not; compare A000977.

T. D. Noe, Table of n, a(n) for n = 1..1000
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

A225228 is a subsequence.
Row 3 of A125666.
Cf. A000977, A000961, A001221, A112802.

a033992 n = a033992_list !! (n-1)
a033992_list = filter ((== 3) . a001221) [1..]
-- Reinhard Zumkeller, May 03 2013

A033992 := proc(n)
if (nops(numtheory[factorset](n)) = 3) then
   RETURN(n)
fi: end:  seq(A033992(n), n=1..500); # Jani Melik, Feb 24 2011

Select[Range[300],PrimeNu[#]==3&] (* Harvey P. Dale, May 01 2013 *)

is(n)=omega(n)==3 \\ Charles R Greathouse IV, Apr 28 2015

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=3)=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

omega(a(n)) = A001221(a(n)) = 3. - Jonathan Vos Post, Sep 20 2005

a(n) ~ 2n log n / (log log n)^2. - Charles R Greathouse IV, Jul 28 2016

A051270 Numbers that are divisible by exactly 5 different primes.

Original entry on oeis.org

2310, 2730, 3570, 3990, 4290, 4620, 4830, 5460, 5610, 6006, 6090, 6270, 6510, 6630, 6930, 7140, 7410, 7590, 7770, 7854, 7980, 8190, 8580, 8610, 8778, 8970, 9030, 9240, 9282, 9570, 9660, 9690, 9870, 10010, 10230, 10374, 10626, 10710, 10920, 11130, 11220, 11310

Offset: 1

Labos Elemer

nonn

			2730 = 2*3*5*7*13 is the first nontrivial 5-prime factor number following the 5th primorial, 2310 = 2*3*5*7*11.

T. D. Noe, Table of n, a(n) for n = 1..1000
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

A046303 is a subsequence.
Row 5 of A125666.
Cf. A000977, A002110.

A051270 := proc(n)
    option remember;
    local a;
    if n = 1 then
        2*3*5*7*11 ;
    else
        for a from procname(n-1)+1 do
            if A001221(a)= 5 then
                return a;
            end if;
        end do:
    end if;
end proc: # R. J. Mathar, Oct 13 2019

Select[Range[12000],PrimeNu[#]==5&] (* Harvey P. Dale, Feb 13 2012 *)

is(n)=omega(n)==5 \\ Charles R Greathouse IV, Apr 29 2015

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=5)=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

from sympy import primefactors
print([n for n in range(2, 20001) if len(primefactors(n))==5]) # Indranil Ghosh, Apr 06 2017

A033993 Numbers that are divisible by exactly four different primes.

Original entry on oeis.org

210, 330, 390, 420, 462, 510, 546, 570, 630, 660, 690, 714, 770, 780, 798, 840, 858, 870, 910, 924, 930, 966, 990, 1020, 1050, 1092, 1110, 1122, 1140, 1155, 1170, 1190, 1218, 1230, 1254, 1260, 1290, 1302, 1320, 1326, 1330, 1365, 1380, 1386, 1410, 1428

Offset: 1

Labos Elemer

For a(n) < 30030 = 2 * 3 * 5 * 7 * 11 * 13 this is identical to "numbers with a semiprime number of distinct prime factors." - Jonathan Vos Post, Sep 21 2005

			The 4th primorial is the first term of this sequence: A002110(4) = 210.

T. D. Noe, Table of n, a(n) for n = 1..1000
Hans Montanus and Ron Westdijk, Cellular Automation and Binomials, Green Blue Mathematics (2022), p. 90.

Row 4 of A125666.

Cf. A000977, A002110, A001221.

Select[Range[1500], Length[FactorInteger[#]] == 4 &] (* Vladimir Joseph Stephan Orlovsky, Apr 22 2010 *)

is(n)=omega(n)==4 \\ Charles R Greathouse IV, Sep 17 2015

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=4)=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

a(n) has exactly 4 distinct prime factors. omega(a(n)) = A001221(a(n)) = 4. - Jonathan Vos Post, Sep 21 2005

A056866 Orders of non-solvable groups, i.e., numbers that are not solvable numbers.

Original entry on oeis.org

60, 120, 168, 180, 240, 300, 336, 360, 420, 480, 504, 540, 600, 660, 672, 720, 780, 840, 900, 960, 1008, 1020, 1080, 1092, 1140, 1176, 1200, 1260, 1320, 1344, 1380, 1440, 1500, 1512, 1560, 1620, 1680, 1740, 1800, 1848, 1860, 1920, 1980, 2016, 2040

Offset: 1

N. J. A. Sloane, Sep 02 2000

A number is solvable if every group of that order is solvable.
This comment is about the three sequences A001034, A060793, A056866: The Feit-Thompson theorem says that a finite group with odd order is solvable, hence all numbers in this sequence are even. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 08 2001 [Corrected by Isaac Saffold, Aug 09 2021]
Insoluble group orders can be derived from A001034 (simple non-cyclic orders): k is an insoluble order iff k is a multiple of a simple non-cyclic order. - Des MacHale
All terms are divisible by 4 and either 3 or 5. - Charles R Greathouse IV, Sep 11 2012
Subsequence of A056868 and hence of A060652. - Charles R Greathouse IV, Apr 16 2015, updated Sep 11 2015
The primitive elements are A257146. Since the sum of the reciprocals of the terms of that sequence converges, this sequence has a natural density and so a(n) ~ k*n for some k (see, e.g., Erdős 1948). - Charles R Greathouse IV, Apr 17 2015
From Jianing Song, Apr 04 2022: (Start)
Burnside's p^a*q^b theorem says that a finite group whose order has at most 2 distinct prime factors is solvable, hence all terms have at least 3 distinct prime factors.
Terms not divisible by 12 are divisible by 320 and have at least 4 distinct prime factors (cf. A257391). (End)

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000 (first 2240 terms from T. D. Noe)
R. Brauer, Investigation on groups of even order, I.
R. Brauer, Investigation on groups of even order, II.
P. Erdős, On the density of some sequences of integers, Bull. Amer. Math. Soc. 54 (1948), pp. 685-692. See p. 685.
W. Feit and J. G. Thompson, A solvability criterion for finite groups and consequences, Proc. N. A. S. 48 (6) (1962) 968.
J. Pakianathan and K. Shankar, Nilpotent numbers, Amer. Math. Monthly, 107, August-September 2000, 631-634.
Cindy Tsang, Qin Chao, On the solvability of regular subgroups in the holomorph of a finite solvable group, arXiv:1901.10636 [math.GR], 2019.
Index entries for sequences related to groups

Subsequence of A000977 and A056868.

Cf. A003277, A051532, A056867, A001034, A060652.

ma[n_] := For[k = 1, True, k++, p = Prime[k]; m = 2^p*(2^(2*p) - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mb[n_] := For[k = 2, True, k++, p = Prime[k]; m = 3^p*((3^(2*p) - 1)/2); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; mc[n_] := For[k = 3, True, k++, p = Prime[k]; m = p*((p^2 - 1)/2); If[Mod[p^2 + 1, 5] == 0, If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]]; md[n_] := Mod[n, 2^4*3^3*13] == 0; me[n_] := For[k = 2, True, k++, p = Prime[k]; m = 2^(2*p)*(2^(2*p) + 1)*(2^p - 1); If[m > n, Return[False], If[Mod[n, m] == 0, Return[True]]]]; notSolvableQ[n_] := OddQ[n] || ma[n] || mb[n] || mc[n] || md[n] || me[n]; Select[ Range[3000], notSolvableQ] (* Jean-François Alcover, Jun 14 2012, from formula *)

is(n)={
    if(n%5616==0,return(1));
    forprime(p=2,valuation(n,2),
        if(n%(4^p-1)==0, return(1))
    );
    forprime(p=3,valuation(n,3),
        if(n%(9^p\2)==0, return(1))
    );
    forprime(p=3,valuation(n,2)\2,
        if(n%((4^p+1)*(2^p-1))==0, return(1))
    );
    my(f=factor(n)[,1]);
    for(i=1,#f,
        if(f[i]>3 && f[i]%5>1 && f[i]%5<4 && n%(f[i]^2\2)==0, return(1))
    );
    0
}; \\ Charles R Greathouse IV, Sep 11 2012

A positive integer k is a non-solvable number if and only if it is a multiple of any of the following numbers: a) 2^p*(2^(2*p)-1), p any prime. b) 3^p*(3^(2*p)-1)/2, p odd prime. c) p*(p^2-1)/2, p prime greater than 3 such that p^2 + 1 == 0 (mod 5). d) 2^4*3^3*13. e) 2^(2*p)*(2^(2*p)+1)*(2^p-1), p odd prime.

More terms from Des MacHale, Feb 19 2001

Further terms from Francisco Salinas (franciscodesalinas(AT)hotmail.com), Dec 25 2001

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.

A335520 Number of (1,2,3)-matching permutations of the prime indices of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 19 2020

nonn

We define a pattern to be a finite sequence covering an initial interval of positive integers. Patterns are counted by A000670 and ranked by A333217. A sequence S is said to match a pattern P if there is a not necessarily contiguous subsequence of S whose parts have the same relative order as P. For example, (3,1,1,3) matches (1,1,2), (2,1,1), and (2,1,2), but avoids (1,2,1), (1,2,2), and (2,2,1).

			The a(n) permutations for n = 30, 60, 120, 210, 180, 480:
  (123)  (1123)  (11123)  (1234)  (11223)  (1111123)
         (1213)  (11213)  (1243)  (11232)  (1111213)
         (1231)  (11231)  (1324)  (12123)  (1111231)
                 (12113)  (1342)  (12132)  (1112113)
                 (12131)  (1423)  (12213)  (1112131)
                 (12311)  (2134)  (12231)  (1112311)
                          (2314)  (12312)  (1121113)
                          (2341)  (12321)  (1121131)
                          (3124)  (21123)  (1121311)
                          (4123)  (21213)  (1123111)
                                  (21231)  (1211113)
                                           (1211131)
                                           (1211311)
                                           (1213111)
                                           (1231111)

Wikipedia, Permutation pattern
Gus Wiseman, Sequences counting and ranking compositions by the patterns they match or avoid.

Positions of nonzero terms are A000977.
These permutations are ranked by A335479.
These compositions are counted by A335514.
Patterns matching this pattern are counted by A335515.
The complement A335521 is the avoiding version.
Permutations of prime indices are counted by A008480.
Patterns are counted by A000670 and ranked by A333217.
Anti-run permutations of prime indices are counted by A335452.
Cf. A056239, A056986, A112798, A238279, A281188, A333221, A333755, A335456, A335460, A335462, A335463.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[Permutations[primeMS[n]],MatchQ[#,{_,x_,_,y_,_,z_,_}/;x

For n > 0, a(n) + A335521(n) = A008480(n).

A070915 Numbers having at most two distinct prime factors.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 61, 62, 63, 64, 65, 67, 68, 69, 71

Offset: 1

Reinhard Zumkeller, May 20 2002

nonn

			88 and 77 are terms, as 88=11*2^3 and 77=11*7, but 66 is not a term, as 66 = 11*3*2.

G. C. Greubel, Table of n, a(n) for n = 1..10000

Disjoint union of A000961 and A007774.
A003586 is a subsequence.
Cf. A001221.
Complement of A000977.

Select[Range[80],PrimeNu[#]<3&] (* Harvey P. Dale, Sep 19 2012 *)

is(n)=omega(n)<3 \\ Charles R Greathouse IV, Oct 16 2015

A001221(a(n)) <= 2.

a(n) ~ n log n/log log n. - Charles R Greathouse IV, Oct 16 2015

cp's OEIS Frontend

A264780 a(n) is the index of A246947(n) in A000977.

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A179938 Third largest prime factor of numbers that are divisible by at least three different primes (A000977).

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A046387 Products of exactly 5 distinct primes.

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A033992 Numbers that are divisible by exactly three different primes.

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A051270 Numbers that are divisible by exactly 5 different primes.

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A033993 Numbers that are divisible by exactly four different primes.

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A056866 Orders of non-solvable groups, i.e., numbers that are not solvable numbers.