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

%I A259350 #66 Jul 20 2015 16:17:46
%S A259350 41704979954,124731595066,365993436094,366230785766,367810728790,
%T A259350 368695198806,589316590786,598986161410,607638803134,673917791834,
%U A259350 710756189898,753389272714,762118572046,772416848554,806996241806,832216749090,874567856590,905173650094,933893335166,958872775134,970959170390,985722818366,997785568130
%N A259350 Numbers n such that n-1, n, and n+1 are all products of 7 distinct primes.
%C A259350 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.
%C A259350 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.
%C A259350 Note: The PARI program avoids duplicates but does not order terms.
%H A259350 Charles R Greathouse IV, <a href="/A259350/b259350.txt">Table of n, a(n) for n = 1..2510</a>
%e A259350 41704979953 = 7*13*29*41*47*59*139,
%e A259350 41704979954 = 2*11*23*31*83*103*311, and
%e A259350 41704979955 = 3*5*17*19*109*157*503; and no smaller such trio exists, so that a(1)=41704979954.
%o A259350 (PARI)
%o A259350 {
%o A259350 \\Program runs for arbitrary B.\\
%o A259350 B=10^12;N=primepi(B/(627*17*19*23));
%o A259350 p=vector(N,n,prime(n));
%o A259350 in=primepi((B/210)^(1/3));
%o A259350 P=prod(i=1,27,p[i]);Q=prod(i=28,in,p[i]);
%o A259350 v=28;d=[[1,2],[-1,1],[-2,-1]];i3=6;
%o A259350 while(6*p[i3]^5<B,
%o A259350 k1=p[i3];
%o A259350 for(i2=2,i3-1,
%o A259350   k1*=p[i2];
%o A259350   for(i1=1,i2-1,
%o A259350    k1*=p[i1];
%o A259350    if(k1>626,
%o A259350     if(k1*p[i3+1]*p[i3+2]*p[i3+3]*p[i3+4]<B,
%o A259350      if(6*p[v-1]<k1,
%o A259350       while(6*p[v]<k1,Q/=p[v];P*=p[v];v++),
%o A259350       while(6*p[v-1]>=k1,v--;Q*=p[v];P/=p[v]));
%o A259350      r=(B\k1)^(1/4);j1=i3+1;
%o A259350      while(p[j1]<r,
%o A259350       k2=p[j1]*k1;s(B\k2)^(1/3);j2=j1+1;
%o A259350       while(p[j2]<s,
%o A259350        k3=p[j2]*k2;t=sqrt(B\k3);j3=j2+1;
%o A259350        while(p[j3]<t,
%o A259350         k4=k3*p[j3];j4=j3+1;u=B\k4;
%o A259350         while(p[j4]<=u,
%o A259350          K=k4*p[j4];
%o A259350          a=K+d[K%4][1];b=K+d[K%4][2];
%o A259350          a1=gcd(P,a);y1=omega(a1);
%o A259350          if(y1>2,
%o A259350           f=1;if(y1==3,if(a1>j1,f=0));
%o A259350           if(f,
%o A259350            b1=gcd(P,b);z1=omega(b1);
%o A259350            if(z1>2,
%o A259350             if(z1==3,if(b1>j1,f=0));
%o A259350             if(f,
%o A259350              a2=a/a1;
%o A259350              if(gcd(a1,a2)==1,
%o A259350               b2=b/b1;
%o A259350               if(gcd(b1,b2)==1,
%o A259350                a21=gcd(a2,Q);a22=a2/a21;
%o A259350                if(gcd(a21,a22)==1,
%o A259350                 y=y1+omega(a21);
%o A259350                 if(y>4,
%o A259350                  if(y<8,
%o A259350                   b21=gcd(Q,b2);b22=b2/b21;
%o A259350                   if(gcd(b21,b22)==1,
%o A259350                    z=z1+omega(b21);
%o A259350                    if(z>4,
%o A259350                     if(z<8,
%o A259350                      if(y+omega(a22)==7,
%o A259350                       if(z+omega(b22)==7,
%o A259350                        f1=factor(a1);
%o A259350                        if(f1[1,1]*f1[2,1]*f1[3,1]<k1,
%o A259350                         f2=factor(b1);
%o A259350                         if(f2[1,1]*f2[2,1]*f2[3,1]<k1,
%o A259350                          print1((K\4)*4+2" ")
%o A259350          ))))))))))))))));
%o A259350         j4++);
%o A259350        j3++);
%o A259350       j2++);
%o A259350      j1++)));
%o A259350   k1/=p[i1]);
%o A259350 k1/=p[i2]);
%o A259350 i3++)
%o A259350 }
%Y A259350 Cf. A093550, A169834, A248201, A248202, A248203, A248204, A248349, A259801.
%K A259350 nonn
%O A259350 1,1
%A A259350 _James G. Merickel_, Jun 24 2015