A000977 -id:A000977 - cp's OEIS frontend

A122181 Numbers k that can be written as k = xyz with 1 < x < y < z (A122180(k) > 0).

24, 30, 36, 40, 42, 48, 54, 56, 60, 64, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 105, 108, 110, 112, 114, 120, 126, 128, 130, 132, 135, 136, 138, 140, 144, 150, 152, 154, 156, 160, 162, 165, 168, 170, 174, 176, 180, 182, 184, 186, 189, 190, 192, 195

Offset: 1

Rick L. Shepherd, Aug 24 2006

Equivalently, numbers k with at least 7 divisors (A000005(k) > 6). Equivalently, numbers k with at least 5 proper divisors (A070824(k) > 4). Equivalently, numbers k such that i) k has at least three distinct prime factors (A000977), ii) k has two distinct prime factors and four or more total prime factors (k = p^j*q^m, p,q primes, j+m >= 4), or iii) k = p^m, a perfect power (A001597) but restricted to prime p and m >= 6 [= 1+2+3] (some terms of A076470).

			a(1) = 24 = 2*3*4, a product of three distinct proper divisors (omega(24) = 2, bigomega(24) = 4).
a(2) = 30 = 2*3*5, a product of three distinct prime factors (omega(30) = 3).
a(10) = 64 = 2*4*8 [= 2^1*2^2*2^3] (omega(64) = 1, bigomega(64) = 6).

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

Cf. A122179, A122180, A000005, A070824, A000977, A001221, A001222, A001597, A076470.

Select[Range[200], DivisorSigma[0, #] > 6 &] (* Amiram Eldar, Oct 05 2024 *)

PARI
```
isok(n) = numdiv(n)>6
```

isok(n) = (omega(n)==1 && bigomega(n)>5) || (omega(n)==2 && bigomega(n)>3) || (omega(n)>2)

A321503 Numbers m such that m and m+1 both have at least 3 distinct prime factors.

Original entry on oeis.org

230, 285, 429, 434, 455, 494, 560, 594, 609, 615, 644, 645, 650, 665, 714, 740, 741, 759, 804, 805, 819, 825, 854, 860, 884, 902, 935, 945, 969, 986, 987, 1001, 1014, 1022, 1034, 1035, 1044, 1064, 1065, 1070, 1085, 1104, 1105, 1130, 1196, 1209, 1220, 1221, 1235, 1239, 1245, 1265

Offset: 1

M. F. Hasler, Nov 13 2018

nonn

Disjoint union of A140077 (omega({m, m+1}) = {3}) and A321493 (not both have exactly 3 prime divisors). The latter contains terms with indices {15, 60, 82, 98, 99, 104, ...} of this sequence.

Numbers m and m+1 can never have a common prime factor (consider them mod p), therefore the terms are > sqrt(A002110(3+3)), A002110 = primorial.

M. F. Hasler, Table of n, a(n) for n = 1..10000

Subsequence of A000977.
Cf. A255346, A321504 .. A321506, A321489 (analog for k = 2, ..., 7 prime divisors).
Cf. A321493, A321494 .. A321497 (subsequences of the above: m or m+1 has more than k prime divisors).
Cf. A074851, A140077, A140078, A140079 (complementary subsequences: m and m+1 have exactly k = 2, 3, 4, 5 prime divisors).

aQ[n_]:=Module[{v={PrimeNu[n], PrimeNu[n+1]}}, Min[v]>2]; Select[Range[1300], aQ] (* Amiram Eldar, Nov 12 2018 *)

select( is(n)=omega(n)>2&&omega(n+1)>2, [1..1300])

a(n) ~ n. - Charles R Greathouse IV, Jan 25 2025

A246947 a(1)=30; for n > 1, a(n) is the least integer not occurring earlier such that a(n) shares exactly three distinct prime divisors with a(n-1).

Original entry on oeis.org

30, 60, 90, 120, 150, 180, 210, 42, 84, 126, 168, 252, 294, 336, 378, 420, 70, 140, 280, 350, 490, 560, 630, 105, 315, 525, 735, 840, 240, 270, 300, 330, 66, 132, 198, 264, 396, 462, 154, 308, 616, 770, 110, 220, 440, 550, 660, 165, 495, 825, 990, 360, 390, 78

Offset: 1

Michel Lagneau, Sep 08 2014

nonn

All terms belong to A000977. Is this a permutation of A000977? - Michel Marcus, Nov 24 2015

			90 is in the sequence because the common prime distinct divisors between a(2)=60 and a(3)=90 are 2, 3 and 5.

Michel Lagneau, Table of n, a(n) for n = 1..2000

Cf. A000977, A064413, A246946.

with(numtheory):a0:={2,3,5}:lst:={}:
for n from 1 to 100 do:
  ii:=0:
    for k from 30 to 50000 while(ii=0) do:
      y:=factorset(k):n0:=nops(y):lst1:={}:
        for j from 1 to n0 do:
        lst1:=lst1 union {y[j]}:
        od:
         a1:=a0 intersect lst1:
         if {k} intersect lst ={} and a1 <> {} and nops(a1)=3
          then
          printf(`%d, `,k):lst:=lst union {k}:a0:=lst1:ii:=1:
         else
         fi:
      od:
  od:

f[s_List]:=Block[{m=s[[-1]],k=30},While[MemberQ[s,k]||Intersection[Transpose[FactorInteger[k]][[1]],Transpose[FactorInteger[m]][[1]]]=={}|| Length[Intersection[Transpose[FactorInteger[k]][[1]],Transpose[FactorInteger[m]][[1]]]]!=3,k++];Append[s,k]];Nest[f,{30},70]

lista(nn) = {a = 30; print1(a, ", "); fa = (factor(a)[,1])~; va = [a]; k = 0; while (k!= nn, k = 1; while (!((#setintersect(fa, (factor(k)[,1])~) == 3) && (! vecsearch(va, k))), k++); a = k; print1(a, ", "); fa = (factor(a)[,1])~; va = vecsort(concat(va, k)););} \\ Michel Marcus, Nov 24 2015

A064040 Integers whose number of distinct prime divisors is prime.

Original entry on oeis.org

6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 60, 62, 63, 65, 66, 68, 69, 70, 72, 74, 75, 76, 77, 78, 80, 82, 84, 85, 86, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100, 102, 104, 105

Offset: 1

Lior Manor, Aug 23 2001

For all terms below 210 this sequence and A024619 are identical.

			210 = 2*3*5*7 has 4 prime factors, hence it is not here, but it is part of A024619.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A001221, A024619, A000977, A033992, A007774, A033993, A051270, A063989.

q:= n-> isprime(nops(ifactors(n)[2])):
select(q, [$1..210])[];  # Alois P. Heinz, Apr 18 2024

Select[Range[200], PrimeQ[PrimeNu[#]] &] (* Paolo Xausa, Mar 28 2024 *)

n=0; for (m=1, 10^9, if (isprime(omega(m)), write("b064040.txt", n++, " ", m); if (n==1000, break))) \\ Harry J. Smith, Sep 06 2009

is(n)=isprime(omega(n)) \\ Charles R Greathouse IV, Sep 18 2015

Edited by Charles R Greathouse IV, Mar 18 2010

Name edited by Michel Marcus, Oct 16 2023

A168626 Numbers n such that n and n+-1 have 3 or more distinct prime factors.

Original entry on oeis.org

645, 741, 805, 987, 1035, 1065, 1105, 1221, 1275, 1309, 1310, 1463, 1495, 1581, 1749, 1885, 1886, 1925, 1989, 2014, 2015, 2109, 2135, 2211, 2255, 2261, 2289, 2295, 2331, 2355, 2365, 2379, 2409, 2465, 2485, 2541, 2584, 2585, 2665, 2666, 2667, 2679, 2685

Offset: 1

Vladimir Joseph Stephan Orlovsky, Dec 01 2009

nonn

Subsequence of A321503 and hence of A000977.

Cf. A140077

f[n_]:=Length[FactorInteger[n]]; lst={};Do[If[f[n]>=3&&f[n-1]>=3&&f[n+1]>=3,AppendTo[lst,n]],{n,7!}];lst
Mean/@SequencePosition[Table[If[PrimeNu[n]>2,1,0],{n,2700}],{1,1,1}] (* The program uses the SequencePosition function from Mathematica version 10 *) (* Harvey P. Dale, Jun 05 2016 *)

is(n)=omega(n)>2 && omega(n-1)>2 && omega(n+1)>2 \\ Charles R Greathouse IV, Jan 25 2025

a(n) ~ n. - Charles R Greathouse IV, Jan 25 2025

Definition clarified by Harvey P. Dale, Jun 05 2016

A246716 Positive numbers that are not the product of (exactly) two distinct primes.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 11, 12, 13, 16, 17, 18, 19, 20, 23, 24, 25, 27, 28, 29, 30, 31, 32, 36, 37, 40, 41, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 56, 59, 60, 61, 63, 64, 66, 67, 68, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 83, 84, 88, 89, 90, 92, 96, 97, 98, 99, 100

Offset: 1

Giuseppe Coppoletta, Nov 01 2014

Non-disjoint union of A100959 and A000961. Disjoint union of A100959 and A001248.
Complement of A006881, then inheriting the "opposite" of the properties of A006881.
a(n+1) - a(n) <= 4 (gap upper bound) - (that is because among four consecutive integers there is always a multiple of 4, then there is a number which is not the product of two distinct primes). E.g., a(26)-a(25) = a(62)-a(61) = 4. Is it true that for any k <= 4 there are infinitely many numbers n such that a(n+1) - a(n) = k?
If r = A006881(n+1) - A006881(n) - 1 > 1, it indicates that there are r terms of (a(j)) starting with j = A006881(n) - n + 1 which are consecutive integers. E.g., A006881(8) - A006881(7) - 1 = 6, so there are 6 consecutive terms in (a(j)), starting with j = A006881(7) - 7 + 1 = 20.

			7 is a term because 7 is prime, so it has only one prime divisor.
8 and 9 are terms because neither of them has two distinct prime divisors.
30 is a term because it is the product of three primes.
But 35 is not a term because it is the product of two distinct primes.

Giuseppe Coppoletta, Table of n, a(n) for n = 1..10000

Cf. A001358, A100959, A006881, A007774, A001221, A001222, A007304, A000977, A001248, A000961.

[n: n in [1..100] | #PrimeDivisors(n) ne 2 or &*[t[2]: t in Factorization(n)] ne 1]; // Bruno Berselli, Nov 12 2014

filter:= n -> map(t -> t[2],ifactors(n)[2]) <> [1,1]:
select(filter, [$1..1000]); # Robert Israel, Nov 02 2014

Select[Range[125], Not[PrimeOmega[#] == PrimeNu[#] == 2] &] (* Alonso del Arte, Nov 03 2014 *)

isok(n) = (omega(n)!=2) || (bigomega(n) != 2); \\ Michel Marcus, Nov 01 2014

from math import isqrt
from sympy import primepi, primerange
def A246716(n):
    def f(x): return int(n-(t:=primepi(s:=isqrt(x)))-(t*(t-1)>>1)+sum(primepi(x//k) for k in primerange(1, s+1)))
    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

def A246716_list(n) :
    R = []
    for i in (1..n) :
        d = prime_divisors(i)
        if len(d) != 2 or d[0]*d[1] != i : R.append(i)
    return R
A246716_list(100)

[n for n in (1..100) if sloane.A001221(n)!=2 or sloane.A001222(n)!=2] # Giuseppe Coppoletta, Jan 19 2015

A378720 a(n) is the numerator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

Original entry on oeis.org

0, 0, 1, 1, 4, 326, 628, 992, 98304, 125568, 733440, 281163264, 386427322368, 3178249003008, 12454223855616, 6450728943845376, 342348724735967232, 20218431581110665216, 39814891891080560640, 82739188294287768944640, 15336676441718784000, 61298453882755419734016000

Offset: 1

Robert G. Wilson v and Amiram Eldar, Dec 05 2024

The third smallest prime divisor of a number k is the third member in the ordered list of the distinct prime divisors of k. Only numbers in A000977 have a third smallest prime divisor.

The partial sums of the fractions first exceed 1/2 after summing 4467 terms. Therefore, the median value of the distribution of the third prime divisor is prime(4467) = 42719 = A284411(3).

			The fractions begin with 0/1, 0/1, 1/30, 1/30, 4/165, 326/15015, 628/36465, 992/62985, 98304/7436429, 125568/11849255, ..., .
a(1) = a(2) = 0 since there are no numbers whose third prime divisor is 2 or 3.
a(3)/A378721(3) = 1/30 since the numbers whose third prime divisor is 5 are the numbers that are divisible by 2, 3 and 5, and their density if (1/2)*(1/3)*(1/5) = 1/30.
a(4)/A378721(4) = 1/30 since the numbers whose third prime divisor is 7 are the union of the numbers that are divisible by 2, 3 and 7 and not by 5 whose density is (1/2)*(1/3)*(1-1/5)*(1/7) = 2/105, the numbers that are divisible by 2, 5 and 7 and not by 3 whose density is (1/2)*(1-1/3)*(1/5)*(1/7) = 1/105, and the numbers that are divisible by 3, 5 and 7 and not by 2 whose density is (1-1/2)*(1/3)*(1/5)*(1/7) = 1/210, and 2/105 + 1/105 + 1/210 = 1/30.

József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, Chapter 4, pp. 337-341.

Amiram Eldar, Table of n, a(n) for n = 1..365
Jean-Marie de Koninck and Gérald Tenenbaum, Sur la loi de répartition du k-ième facteur premier d'un entier, Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 133, No. 2 (2002), pp. 191-204.
Paul Erdős and Gérald Tenenbaum, Sur les densités de certaines suites d'entiers, Proc. London Math. Soc. (3), Vol. 59, No. 3 (1989), pp. 417-438; alternative link.

Cf. A000040, A002110, A000977, A038110, A038111, A284411, A342479, A342480, A378721 (denominators).

a[n_] := Block[{p, q = Prime@ Range@ n}, p = Fold[Times, 1, q]; q = Most@ q; Plus @@ Times @@@ Subsets[q -1, {n -3}]/p]; a[1] = 0; Numerator@ Array[a, 22]

a(n) = {my(v = primes(n), q = vecextract(apply(x -> x-1, v),"^-1"), p = vecprod(v), prd = vecprod(q)/p, sm = 0, sb); forsubset([#q, 2], s, sb = vecextract(q, s); sm += 1/vecprod(sb)); numerator(prd * sm);}

a(n)/A378721(n) = (1/prime(n)#) * (Product_{k=1..n-1} (prime(k) - 1)) * Sum_{j=1..n-1, i=1..j-1} 1/((prime(i)-1)*(prime(j)-1)), where prime(n)# = A002110(n) is the n-th primorial number.
Sum_{n>=1} a(n)/A378721(n) = 1.
Sum_{n=1..m} a(n)/A378721(n) > 1/2 for m >= 4467 = primepi(A284411(3)).

A378885 Numbers that are divisible by at least three different primes and the smallest three of them are consecutive primes.

Original entry on oeis.org

30, 60, 90, 105, 120, 150, 180, 210, 240, 270, 300, 315, 330, 360, 385, 390, 420, 450, 480, 510, 525, 540, 570, 600, 630, 660, 690, 720, 735, 750, 780, 810, 840, 870, 900, 930, 945, 960, 990, 1001, 1020, 1050, 1080, 1110, 1140, 1155, 1170, 1200, 1230, 1260, 1290

Offset: 1

Amiram Eldar, Dec 09 2024

All the positive multiples of 30 (A249674 \ {0}) are terms.
Numbers k such that A151800(A020639(k)) | k and also A101300(A020639(k)) | k.
The asymptotic density of this sequence is Sum_{k>=1} (Product_{j=1..k-1} (1-1/prime(j))) / (prime(k)*prime(k+1)*prime(k+2)) = 0.03943839735407432193784... .

			60 = 2^2 * 3 * 5 is a term since 2, 3 and 5 are consecutive primes.
770 = 2 * 5 * 7 * 11 is not a term since its smallest prime divisor is 2 and it is not divisible by 3, the prime next to 2.
1365 = 3 * 5 * 7 * 13 is a term since 3, 5 and 7 are consecutive primes.

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

Subsequence of A000977.
Subsequences: A046301, A378884.
Cf. A020639, A101300, A151800, A249674.

q[k_] := Module[{p = FactorInteger[k][[;; , 1]]}, Length[p] > 2 && p[[2]] == NextPrime[p[[1]]] && p[[3]] == NextPrime[p[[2]]]]; Select[Range[1300], q]

is(k) = if(k == 1, 0, my(p = factor(k)[,1]); #p > 2 && p[2] == nextprime(p[1]+1) && p[3] == nextprime(p[2]+1));

A175785 Numbers n such that the number of distinct prime divisors of n does not divide phi(n).

Original entry on oeis.org

30, 60, 66, 102, 110, 120, 132, 138, 150, 165, 170, 174, 204, 220, 230, 240, 246, 255, 264, 276, 282, 290, 300, 318, 340, 345, 348, 354, 374, 408, 410, 426, 435, 440, 460, 470, 480, 492, 498, 506, 528, 530, 534, 550, 552, 561, 564, 580, 590, 600, 606, 615

Offset: 1

Enrique Pérez Herrero, Sep 04 2010

nonn

a(n) gives the integers where omega(n) = A001221(n) does not divide phi(n) = A000010(n).

This sequence does not contain any prime powers (A000961), nor any numbers with only two distinct prime divisors (A007774); so it is a subsequence of A000977.

			30 is in this sequence because omega(30)=3 does not divide phi(30)=8.

Enrique Pérez Herrero, Table of n, a(n) for n = 1..10000

Cf. A001221, A000010, A075592, A109425, A109427, A192218, A000977, A000961, A007774.

Select[Range[2,700],Mod[EulerPhi[#],PrimeNu[#]]!=0&] (* Harvey P. Dale, Dec 29 2019 *)

isok(n) = (eulerphi(n) % omega(n) != 0) \\ Michel Marcus, Jun 12 2013

A257891 Numbers that are products of at least three consecutive primes.

Original entry on oeis.org

30, 105, 210, 385, 1001, 1155, 2310, 2431, 4199, 5005, 7429, 12673, 15015, 17017, 20677, 30030, 33263, 46189, 47027, 65231, 82861, 85085, 96577, 107113, 146969, 190747, 215441, 241133, 255255, 290177, 323323, 347261, 392863, 409457, 478661, 510510, 583573

Offset: 1

Reinhard Zumkeller, May 12 2015

nonn

			a(5) = 1001 = 7 * 11 * 13;
a(6) = 1155 = 3 * 5 * 7 * 11;
a(7) = 2310 = 2 * 3 * 5 * 7 * 11;
a(8) = 2431 = 11 * 13 * 17.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000

Cf. A151800, A097889, A000977, A046301 (subsequence).

import Data.Set (singleton, deleteFindMin, insert)
a257891 n = a257891_list !! (n-1)
a257891_list = f $ singleton (30, 2, 5) where
   f s = y : f (insert (w, p, q') $ insert (w `div` p, a151800 p, q') s')
         where w = y * q'; q' = a151800 q
               ((y, p, q), s') = deleteFindMin s

Select[Module[{nn=1000},Flatten[Table[Times@@@Partition[Prime[Range[nn]],d,1],{d,3,7}]]]//Union,#<10^7&] (* Harvey P. Dale, Aug 04 2024 *)

cp's OEIS Frontend

A122181 Numbers k that can be written as k = x*y*z with 1 < x < y < z (A122180(k) > 0).

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A321503 Numbers m such that m and m+1 both have at least 3 distinct prime factors.

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A246947 a(1)=30; for n > 1, a(n) is the least integer not occurring earlier such that a(n) shares exactly three distinct prime divisors with a(n-1).

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A064040 Integers whose number of distinct prime divisors is prime.

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A168626 Numbers n such that n and n+-1 have 3 or more distinct prime factors.

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A246716 Positive numbers that are not the product of (exactly) two distinct primes.

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A378720 a(n) is the numerator of the asymptotic density of numbers whose third smallest prime divisor is prime(n).

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A122181 Numbers k that can be written as k = xyz with 1 < x < y < z (A122180(k) > 0).