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

3, 15, 182, 3135, 81345, 2733549, 114535239, 5708795638, 342503171205, 23622238184727, 1793796113720611, 154171427826650234, 15259854588449248245, 1601991507050573600715, 177792170427340904920562, 22932432920285320919637458, 3164592660873444717893657954, 474678349195034622662784926530

Offset: 1

Author

James G. Merickel, Jul 15 2015

Keywords

• nonn

Comments

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).

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.

Examples

			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.
		

Links

• James G. Merickel, Table of n, a(n) for n = 1..18

Crossrefs

Cf. A060796, A260075.

Programs

PARI

{
p=vector(50,n,prime(n));i=1;
while(1,
  P=prod(j=1,2*i,p[j]);Q=sqrt(P);r=10^1000;
  m=prod(j=1,i,p[j]);a=vectorsmall(2*i-1);
  for(j=1,i,a[j]=1);b=vectorsmall(2*i-1);
  for(j=i,2*i-1,b[j]=1);while(1,
    if(m>Q,if(m