A248202 -id:A248202 - cp's OEIS frontend

A093550 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 is a product of n distinct primes.

34, 1310, 203434, 16467034, 1990586014, 41704979954, 102099792179230, 22192526378762466

Offset: 2

Farideh Firoozbakht, Apr 07 2004, corrected Aug 26 2006

Each term of this sequence is of the form 4k+2.

			a(5)=16467034 because each of the three numbers 16467034-1, 16467034 & 16467034+1 are products of 5 distinct primes (16467033=3*11*17*149*197, 16467034=2*19*23*83*227, 16467035=5*13*37*41*167) and 16467034 is the smallest such number.

Jacques Tramu, Puzzle 371

Cf. A093549, A052215, A093548, A248201, A248202, A248203, A248204, A259349, A259350, A259801.

a[n_] := a[n] = (For[m=1, !(Length[FactorInteger[4m+1]]==n && SquareFreeQ[4m+1] && Length[FactorInteger[4m+2]]==n && SquareFreeQ[4m+2] && Length[FactorInteger[4m+3]]==n && SquareFreeQ[4m+3]), m++ ]; 4m+2); Table[Print[a[n]]; a[n], {n, 2, 6}] (* updated by Jean-François Alcover, Jul 04 2013 *)

a(n)={my(m=1);while(!(issquarefree(m-1)&&issquarefree(m)&&issquarefree(m+1)&&omega(m-1)==n&&omega(m)==n&&omega(m+1)==n),m++);return(m);} main(size)={my(n);return(vector(size,n,a(n+1)));} /* Anders Hellström, Jul 14 2015 */

a(7) added from Jacques Tramu's web site by Farideh Firoozbakht, Aug 26 2006
a(8) from Donovan Johnson, Oct 27 2008
a(9) from James G. Merickel, Jul 24 2015

A248201 Numbers n such that n-1, n and n+1 are all squarefree semiprimes.

Original entry on oeis.org

34, 86, 94, 142, 202, 214, 218, 302, 394, 446, 634, 698, 922, 1042, 1138, 1262, 1346, 1402, 1642, 1762, 1838, 1894, 1942, 1982, 2102, 2182, 2218, 2306, 2362, 2434, 2462, 2518, 2642, 2722, 2734, 3098, 3386, 3602, 3694, 3866, 3902, 3958, 4286, 4414

Offset: 1

James G. Merickel, Oct 03 2014

nonn

A subsequence of A169834. - M. F. Hasler, Oct 26 2014

			33, 34 and 35 factor as 3*11, 2*17 and 5*7, respectively.  No smaller such trio exists, so a(1)=34.

Giovanni Resta, Table of n, a(n) for n = 1..10000
Wikipedia, Semiprime

Cf. A006881, A039833, A169834, A195685, A248202, A248203, A248204, A259349, A259350, A259801.

lst={}; Do[z=n^3 + 3 n^2 + 2 n; If[PrimeOmega[z/n]==PrimeOmega[z/(n + 2)]==4 && PrimeNu[z]==6, AppendTo[lst, n + 1]], {n, 1, 6000, 2}]; lst (* Vincenzo Librandi, Jul 24 2015 *)
SequencePosition[Table[If[SquareFreeQ[n]&&PrimeOmega[n]==2,1,0],{n,4500}],{1,1,1}][[All,1]]+1 (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 11 2018 *)

sq(n)=bigomega(n)==2 && omega(n)==2;
for(n=3,10^4,if(sq(n-1)&&sq(n)&&sq(n+1),print1(n,", ")));
\\ Joerg Arndt, Oct 18 2014

a(n) = A039833(n) + 1. - Michel Marcus, Oct 25 2014

a(n) = 2 * A195685(n). - Torlach Rush, Jun 25 2021

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

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).

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A093550 a(n) is the smallest number m such that each of the numbers m-1, m and m+1 is a product of n distinct primes.

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A248201 Numbers n such that n-1, n and n+1 are all squarefree semiprimes.

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

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

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