A093550 -id:A093550 - cp's OEIS frontend

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

2, 3, 61, 193, 15121, 838561, 807905281, 19896463921, 3059220303001, 3931520917431241

Offset: 1

Rick L. Shepherd, Jun 30 2002 and Jens Kruse Andersen, Jul 28 2002

By definition, each term of this sequence is prime.
a(11) <= 1452591346605212407096281241 (Frederick Schneider), see primepuzzles link. - sent by amd64(AT)vipmail.hu, Dec 21 2007
Prime factors are counted with multiplicity. - Harvey P. Dale, Mar 09 2021

			a(3)=61 because 61 (prime), 62 (=2*31), 63 (=3*3*7) have exactly 1, 2, 3 prime factors respectively, and this is the smallest solution;
a(6)=807905281: 807905281 is prime; 807905281+1=2*403952641;
807905281+2=3*15733*17117; 807905281+3=2*2*1871*107951;
807905281+4=5*11*43*211*1619; 807905281+5=2*3*3*3*37*404357;
807905281+6=7*7*7*7*29*41*283; 807905281 is the smallest number m such that m+k is product of k+1 primes for k=0,1,2,3,4,5,6.

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 61, p. 22, Ellipses, Paris 2008.

alt.math.recreational thread, Consecutive numbers with counting prime factors.
Carlos Rivera, Puzzle 425. Consecutive numbers, increasing quantity of prime factors, The Prime Puzzles & Problems Connection.

Cf. A001222, A093552, A093550, A086560, A124592.

a(1) = A000040(1), a(2) = A005383(1), a(3) = A112998(1), a(4) = A113000(1), a(5) = A113008(1), a(6) = A113150(1).

(* This program is not suitable to compute a large number of terms. *) nmax = 6; kmax = 10^6; a[1] = 2; a[n_] := a[n] = For[k = a[n-1]+n-1, k <= kmax, k++, If[AllTrue[Range[0, n-1], PrimeOmega[k+#] == #+1&], Return[k] ] ]; Table[Print["a(", n, ") = ", a[n]]; a[n], {n, 1, nmax}] (* Jean-François Alcover, Sep 06 2017 *)

a(7) found by Mark W. Lewis
a(8) and a(9) found by Jens Kruse Andersen
a(10) found by Jens Kruse Andersen; probably a(11) > 10^20. - Aug 24 2002
Entry revised by N. J. A. Sloane, Jan 26 2007
Cross-references and editing by Charles R Greathouse IV, Apr 20 2010

A093548 a(n) is the smallest number m such that each of the numbers m and m+1 has n distinct prime divisors.

Original entry on oeis.org

2, 14, 230, 7314, 254540, 11243154, 965009045, 65893166030, 5702759516090, 490005293940084, 76622240600506314

Offset: 1

Farideh Firoozbakht, Apr 06 2004

Prime factors may be repeated in m and m+1. The difference between this sequence and A052215 is that in the latter, no prime factor may be repeated. So A052215 imposes more stringent conditions, hence a(n) <= A052215(n). - N. J. A. Sloane, Nov 21 2015
2^63 < a(12) <= 22593106657425552170. - Donovan Johnson, Jan 08 2009
A115186(n) <= a(n) <= A052215(n). - Zak Seidov, Jan 16 2015

			a(5) = 254540 because 254540=2^2*5*11*13*89; 254541=3*7*17*23*31
and 254540 is the smallest number m which each of the numbers m & m+1 has 5 distinct prime divisors.
In contrast, A052215(5) = 378014 > 254540. - _N. J. A. Sloane_, Nov 21 2015

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 230, p. 65, Ellipses, Paris 2008.

Cf. A052215 (another version), A093549, A093550, A115186.

a[n_] := (For[m=1, !(Length[FactorInteger[m]]==n && Length[FactorInteger[m+1]]==n), m++ ];m);Do[Print[a[n]], {n, 7}]
Flatten[Table[SequencePosition[PrimeNu[Range[260000]],{n,n},1],{n,5}],1][[;;,1]] (* To generate more terms, increase the Range and n constants. *) (* Harvey P. Dale, Jun 08 2023 *)

from sympy import primefactors, primorial
def a(n):
  m = primorial(n)
  while True:
    if len(primefactors(m)) == n:
      if len(primefactors(m+1)) == n: return m
      else: m += 2
    else: m += 1
for n in range(1, 6):
  print(a(n), end=", ") # Michael S. Branicky, Feb 14 2021

a[n_] := (For[m=1, !(Length[FactorInteger[m]]==n && Length[FactorInteger[m+1]]==n), m++ ];m)

a(8), a(9) from Martin Fuller, Jan 17 2006

a(10)-a(11) from Donovan Johnson, Jan 08 2009

A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

Original entry on oeis.org

2, 14, 230, 7314, 378014, 11243154, 965009045, 65893166030, 5702759516090, 605247139068494, 78971815814237709, 22593106657425552170

Offset: 1

Erich Friedman, Jan 29 2000

Prime factors may not be repeated in m and m+1. The difference between this sequence and A093548 is that in the latter, prime factors may be repeated. So the present sequence imposes more stringent conditions than A093548, hence a(n) >= A093548(n). - N. J. A. Sloane, Nov 21 2015
A115186(n) <= A093548(n) <= a(n). - Zak Seidov, Jan 16 2015
2^63 < a(12) <= 22593106657425552170. [Donovan Johnson, Oct 23 2008]
a(12) confirmed to be the upper limit of the range above. - Bert Dobbelaere, Jun 27 2019

			14 and 15 are both the product of 2 primes.
230 is the 3rd entry because we have (230=2*5*23, 231=3*7*11).

Cf. A093548 (another version), A093549, A093550, A115186, A318896.

Subsequence of A005117.

More terms from Naohiro Nomoto, Jul 08 2001
a(7) from Farideh Firoozbakht, Apr 06 2004
a(8)-a(10) from Martin Fuller, Jan 17 2006
a(11) from Donovan Johnson, Oct 23 2008
a(12) from Bert Dobbelaere, Jun 27 2019

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

A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

Original entry on oeis.org

2, 33, 1309, 27962, 3323705, 296602730, 41704979953

Offset: 1

M. F. Hasler, May 18 2014

nonn

As the example of a(4)=27962 shows, "consecutive squarefree numbers" means consecutive elements of A005117, not necessarily consecutive integers that (additionally) are squarefree; this would be a more restrictive condition.

a(8) <= 102099792179229 because A093550 - 1 is an upper bound of the present sequence.

			The two squarefree numbers following a(4)=27962, namely, 27965 and 27966, also have 4 prime divisors just as a(4).

Daniel C. Mayer, Define an "m-triple" to consist of three consecutive squarefree positive integers, each with exactly m prime divisors, Number Theory group on LinkedIn.com

See A242605-A242608 for triples of consecutive squarefree numbers with m=2,..,5 prime factors.
See A246470 for the quadruplet and A246548 for the 5-tuple versions of this sequence.
See A039833, A066509, A176167 and A192203 for triples of consecutive numbers which are squarefree and have m=2,..,5 prime factors.

Edited and a(6)-a(7) added by Hans Havermann, Aug 27 2014

A248203 Numbers n such that n-1, n, and n+1 are the product of 4 distinct primes.

Original entry on oeis.org

203434, 214490, 225070, 258014, 294594, 313054, 315722, 352886, 389390, 409354, 418846, 421630, 452354, 464386, 478906, 485134, 500906, 508046, 508990, 526030, 528410, 538746, 542270, 542794, 548302, 556870, 559690, 569066, 571234, 579886, 582406, 588730

Offset: 1

James G. Merickel, Oct 28 2014

nonn

A subsequence of A066509 and offset by one from A176167.

			203433 factors as 3*19*43*83, 203434 factors as 2*7*11*1321 and 203435 factors as 5*23*29*61; and with no similar smaller trio a(1)=203434. [Corrected by _James G. Merickel_, Jul 23 2015]

Anders Hellström, Table of n, a(n) for n = 1..300

Cf. A066509, A176167, A248201, A248202, A248204, A259349, A259350, A259801.

f1[n_]:=Last/@FactorInteger[n]=={1, 1, 1, 1}; f2[n_]:=Max[Last/@FactorInteger[n]]; lst={}; Do[If[f1[n]&&f1[n + 1]&&f1[n+2], AppendTo[lst, n + 1]], {n, 2 8!, 4 9!}]; lst (* Vincenzo Librandi, Aug 02 2015 *)

{
\\ Initialized at A093550(4) (3rd term there, w/offset=2). If this \\
\\ program is to run from a different starting value of n, it must not \\
\\ be congruent to -1, 0 or 1 modulo 9 (in addition to being congruent \\
\\ to 2 modulo 4), and either u or the vector s needs to be brought into \\
\\ agreement. \\
n=203434;s=[4,4,8,8,8,4];u=1;
while(1,
  if(issquarefree(n) &&
    issquarefree(n-1) &&
    issquarefree(n+1) &&
    omega(n)==4 &&
    omega(n-1)==4 &&
    omega(n+1)==4,
    print1(n, ", "));
  n+=s[u];if(u==6,u=1,u++))
} \\ James G. Merickel, Jul 23 2015

is_ok(n)=(n>1&&omega(n-1)==4&&omega(n)==4&&omega(n+1)==4&&issquarefree(n-1)&&issquarefree(n)&&issquarefree(n+1));
first(m)=my(v=vector(m),i,t=2);for(i=1,m,while(!is_ok(t),t++);v[i]=t;t++);v; /* Anders Hellström, Aug 01 2015 */

a(n) = A176167(n)+1.

A248204 Middle values in trios of products of 5 distinct primes.

Original entry on oeis.org

16467034, 18185870, 21134554, 21374354, 21871366, 22247554, 22412534, 22721586, 24845314, 25118094, 25228930, 25435334, 25596934, 26217246, 27140114, 29218630, 29752346, 30323734, 30563246, 31943066, 32663266, 33367894, 36055046, 38269022, 39738062, 40547066

Offset: 1

James G. Merickel, Oct 28 2014

nonn

A subsequence of A066509 and offset by 1 from A192203.

			16467033=3*11*17*149*197,
16467034=2*19*23*83*227, and
16467035=5*13*37*41*167,
with no smaller similar trio. So a(1)=16467034. [Corrected by _James G. Merickel_, Jul 23 2015]

James G. Merickel, Table of n, a(n) for n = 1..26

Cf. A066509, A192203, A248201, A248202, A248203, A259349, A259350, A259801.

{
\\ This program checks all consecutives with elements not divisible \\
\\ by the squares of 2 or 3. More efficiency is required if enormous \\
\\ numbers of terms are sought and for the analog sequences beyond \\
\\ 6 prime factors. The start value is A093550(5). If a start other \\
\\ than this is chosen, one must be sure that (one of) s or u is \\
\\ adjusted if it needs to be. \\
n=16467034;s=[8,4,4,4,8,8];u=1;
while(1,
  if(issquarefree(n) && issquarefree(n-1) && issquarefree(n+1) && omega(n)==5 && omega(n-1)==5 && omega(n+1)==5, print1(n" "));
  n+=s[u];
  if(u==6,u=1,u++)
)
} \\ James G. Merickel, Jul 23 2015

a(n) = A192203(n) + 1.

A259350 Numbers n such that n-1, n, and n+1 are all products of 7 distinct primes.

Original entry on oeis.org

41704979954, 124731595066, 365993436094, 366230785766, 367810728790, 368695198806, 589316590786, 598986161410, 607638803134, 673917791834, 710756189898, 753389272714, 762118572046, 772416848554, 806996241806, 832216749090, 874567856590, 905173650094, 933893335166, 958872775134, 970959170390, 985722818366, 997785568130

Offset: 1

James G. Merickel, Jun 24 2015

nonn

A subsequence of A169834. A093550(7)=a(1), that sequence with offset 2 (so actually its 6th term) holding first terms of sequences of this kind.
Other than a(4)=366230785766 and a(18)=905173650094 (with minimax prime factor 1867 for it and its neighbors), the terms were initially discovered by increasing value of the trios' smallest large prime factors. An exhaustive search running multiple (suitably modified) copies of a pre-acceptance PARI program that disposed of fails in a somewhat efficient way and ran about an order of magnitude faster than the analog of the simple program at A259349 required about 1000 window-hours to produce the list given (adding two terms, including one that was unachievable by the increasing-minimax-prime method). Then the much faster program--15 minutes in just one PARI window--shown was developed and edited in here in its place. By specifying the 4 largest prime factors secondary to setting the product of the smallest 3 such that this is at least 627--as must be true for one of 3 relatively prime sphenic numbers--a speedup of over 3.5 orders of magnitude more (over the single order of magnitude that the replaced program managed, for a total of about 10^4.5 in time ratio over the program used for 6 primes) was achieved.
Note: The PARI program avoids duplicates but does not order terms.

			41704979953 = 7*13*29*41*47*59*139,
41704979954 = 2*11*23*31*83*103*311, and
41704979955 = 3*5*17*19*109*157*503; and no smaller such trio exists, so that a(1)=41704979954.

Charles R Greathouse IV, Table of n, a(n) for n = 1..2510

Cf. A093550, A169834, A248201, A248202, A248203, A248204, A248349, A259801.

{
\\Program runs for arbitrary B.\\
B=10^12;N=primepi(B/(627*17*19*23));
p=vector(N,n,prime(n));
in=primepi((B/210)^(1/3));
P=prod(i=1,27,p[i]);Q=prod(i=28,in,p[i]);
v=28;d=[[1,2],[-1,1],[-2,-1]];i3=6;
while(6*p[i3]^5626,
    if(k1*p[i3+1]*p[i3+2]*p[i3+3]*p[i3+4]=k1,v--;Q*=p[v];P/=p[v]));
     r=(B\k1)^(1/4);j1=i3+1;
     while(p[j1]2,
          f=1;if(y1==3,if(a1>j1,f=0));
          if(f,
           b1=gcd(P,b);z1=omega(b1);
           if(z1>2,
            if(z1==3,if(b1>j1,f=0));
            if(f,
             a2=a/a1;
             if(gcd(a1,a2)==1,
              b2=b/b1;
              if(gcd(b1,b2)==1,
               a21=gcd(a2,Q);a22=a2/a21;
               if(gcd(a21,a22)==1,
                y=y1+omega(a21);
                if(y>4,
                 if(y<8,
                  b21=gcd(Q,b2);b22=b2/b21;
                  if(gcd(b21,b22)==1,
                   z=z1+omega(b21);
                   if(z>4,
                    if(z<8,
                     if(y+omega(a22)==7,
                      if(z+omega(b22)==7,
                       f1=factor(a1);
                       if(f1[1,1]*f1[2,1]*f1[3,1]

A259801 Numbers such that it and its two neighbors are products of 8 distinct primes.

Original entry on oeis.org

102099792179230, 117092756174954, 136745109677256, 162338633743714, 167791215874866, 178571623400554, 183789996331514, 188284244083286, 211843056257854, 217181576415166, 224685381821406, 230455538364206, 234115003437666, 247662164889294, 265223112108514, 265730468260830, 266665427846390, 267248859559214, 268021718391414, 274354628059534

Offset: 1

James G. Merickel, Jul 14 2015

nonn

A subsequence of A169834.
With bound set at 4*10^14, the linked-to PARI program completed its run in about 2 days (producing 48 terms). The program fixes prospective smallest 4 prime factors so their product is at or above the minimum possible of the largest of 3 products of 4 primes without overlap (A260075(4)=20553), doing bound-restricted testing for the larger 4 in turn for each of these smaller quadruples. This is just one of a variety of ways of fixing a prospective trio by specifying one member as being within a certain range and satisfying the criterion. The program mostly avoids duplicates but does not entirely. See the part of the corresponding program at A259350 immediately before the print command for a fix.
The efficiency the program seems to generate empirically would come from the specification of product of 4 smaller primes as greater than a certain value and whole product within a certain range. Running through all even products of 8 distinct primes between the cube root of the (3n)-th primorial and the bound given would be a simpler way but one not so statistically limited (with a proportionally larger number of candidates). Note: The author is not making a claim of maximal efficiency, just of marked improvements over some simpler approaches.
a(1)=A093550(8).

			102099792179229=3*13*19*53*83*131*181*1321, 102099792179230=2*5*17*43*127*229*283*1697, and 102099792179231=7*11*23*29*31*71*113*7993. No smaller collection meets the criterion, so a(1)=102099792179230.

James G. Merickel, Table of n, a(n) for n = 1..48
James G. Merickel, PARI program

Cf. A093550, A169834, A248201, A248202, A248203, A248204, A259349, A259350, A260075.

\\ See above link to PARI program generating terms under 4*10^14 (out of order and with some duplicates).

A113752 Smallest number m such that m, m+1 and m+2 have exactly n prime factors (counted with multiplicity).

Original entry on oeis.org

33, 170, 1274, 15470, 33614, 3145310, 40909374, 668363967, 9864741248, 179199427328, 967461818750, 57938945781248, 597779386906624

Offset: 2

Martin Fuller, Jan 17 2006

n = A001222(a(n)) = A001222(a(n)+1) = A001222(a(n)+2). Subsequence of A045920.
a(14) <= 1247579465781248. - Donovan Johnson, Jun 12 2013
a(15) > 2 * 10^15. - Toshitaka Suzuki, Aug 31 2025

			a(6) = 33614 = 2*7*7*7*7*7, a(6)+1 = 3*3*3*3*5*83, a(6)+2 = 2*2*2*2*11*191

Cf. A001222, A045920, A093549, A093550, A115186.

t = {}; n = 2; m = 1; While[Length[t] < 5, m++; If[n == PrimeOmega[m] == PrimeOmega[m + 1] == PrimeOmega[m + 2], AppendTo[t, m]; n++]]; t (* T. D. Noe, Aug 19 2013 *)

a(9)-a(11) from Donovan Johnson, Apr 08 2008
a(12) from Donovan Johnson, Aug 08 2011
a(13) from Jud McCranie, Aug 19 2013
a(14) from Toshitaka Suzuki, Aug 31 2025

cp's OEIS Frontend

A072875 Smallest start for a run of n consecutive numbers of which the i-th has exactly i prime factors.

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A093548 a(n) is the smallest number m such that each of the numbers m and m+1 has n distinct prime divisors.

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A052215 a(n) = smallest number m such that m and m+1 are the product of exactly n distinct primes.

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

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A242621 Start of the least triple of consecutive squarefree numbers each of which has exactly n distinct prime factors.

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A248203 Numbers n such that n-1, n, and n+1 are the product of 4 distinct primes.

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Mathematica

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A248204 Middle values in trios of products of 5 distinct primes.

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A259350 Numbers n such that n-1, n, and n+1 are all products of 7 distinct primes.

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