cp's OEIS Frontend

A260079 The smallest product of n distinct primes no larger than the (2n)-th to exceed the square root of the (2n)-th primorial.

%I A260079 #28 Jul 23 2015 17:28:55
%S A260079 3,15,182,3135,81345,2733549,114535239,5708795638,342503171205,
%T A260079 23622238184727,1793796113720611,154171427826650234,
%U A260079 15259854588449248245,1601991507050573600715,177792170427340904920562,22932432920285320919637458,3164592660873444717893657954,474678349195034622662784926530
%N A260079 The smallest product of n distinct primes no larger than the (2n)-th to exceed the square root of the (2n)-th primorial.
%C A260079 This is similar to but distinct from the even-indexed terms of A060796, with a(n) differing from A060796(2n) at n=7, 10, 11, 12, 13 and 16 (with A060796(36) unavailable for comparison). A260075 is the analog by splitting the first 3n primes into 3 equal-sized sets (but not by giving the smallest product larger than the cube root of the corresponding primorial). The percentages by which a(n) exceeds the square root of the (2n)-th primorial are 22.5, 3.51, 5.03, 0.660, 1.13, 0.347, 0.136, 1.82*10^(-3), 8.54*10^(-3), 6.21*10^(-3), 9.28*10^(-4), 1.84*10^(-4), 1.71*10^(-4), 1.31*10^(-5), 1.94*10^(-6), 5.62*10^(-8), 2.93*10^(-7) and 4.50*10^(-8).
%C A260079 The below PARI program functions by checking for each set of n primes through the (2n-1)-st whether either its product or its product's cofactor in the (2n)-th primorial gives an improvement.
%H A260079 James G. Merickel, <a href="/A260079/b260079.txt">Table of n, a(n) for n = 1..18</a>
%e A260079 If the primes 2 through 13 are split into equal-sized sets, the larger product of the two is not less than 182. This is the value for {2, 7, 13}, so a(3)=182.
%o A260079 (PARI)
%o A260079 {
%o A260079 p=vector(50,n,prime(n));i=1;
%o A260079 while(1,
%o A260079   P=prod(j=1,2*i,p[j]);Q=sqrt(P);r=10^1000;
%o A260079   m=prod(j=1,i,p[j]);a=vectorsmall(2*i-1);
%o A260079   for(j=1,i,a[j]=1);b=vectorsmall(2*i-1);
%o A260079   for(j=i,2*i-1,b[j]=1);while(1,
%o A260079     if(m>Q,if(m<r,r=m),if(P/m<r,r=P/m));
%o A260079     if(a==b,
%o A260079       print1(r" ");break(),
%o A260079       k=1;if(a[1]==0,
%o A260079         while(a[k]==0,k++);c=1;
%o A260079         while(a[k+1],m*=p[c]/p[k];a[c]=1;a[k]=0;c++;k++);
%o A260079         a[k]=0;a[k+1]=1;m*=p[k+1]/p[k],
%o A260079         while(a[k+1],k++);a[k]=0;a[k+1]=1;m*=p[k+1]/p[k])));
%o A260079   i++)
%o A260079 }
%Y A260079 Cf. A060796, A260075.
%K A260079 nonn
%O A260079 1,1
%A A260079 _James G. Merickel_, Jul 15 2015