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

1,1

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.

Links

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

Example

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.

Prog

(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<r, r=m), if(P/m<r, r=P/m));
    if(a==b,
      print1(r" "); break(),
      k=1; if(a[1]==0,
        while(a[k]==0, k++); c=1;
        while(a[k+1], m*=p[c]/p[k]; a[c]=1; a[k]=0; c++; k++);
        a[k]=0; a[k+1]=1; m*=p[k+1]/p[k],
        while(a[k+1], k++); a[k]=0; a[k+1]=1; m*=p[k+1]/p[k])));
  i++)
}

Crossrefs

Cf. A060796, A260075.

Sequence in context: A153079 A173301 A361053 * A363506 A364075 A087614

Adjacent sequences: A260076 A260077 A260078 * A260080 A260081 A260082

Keyword

nonn

Author

James G. Merickel, Jul 15 2015

Status

approved