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A260075 Where the first 3n primes are partitioned into 3 sets of cardinality n, the minimum of the largest product. 3
5, 35, 627, 20553, 859066, 48993082, 3441790495, 287535325407, 28839054633794, 3161858853009549, 416108939893639594, 60850811089314245258, 9874934149007840709407, 1754123227439445139773155 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Distinct from the smallest product of n primes from the first 3n that is larger than the cube root of the (3n)-th primorial. Both this sequence and that (hypothetical) one are analogs of A260079.

The below PARI program runs through each product of n primes up to the (3n)-th, testing it for whether it is both greater than the cube root of the (3n)-th primorial and less than the smallest result to that point; and, if these limitations are met, it then goes on to determine whether or not the remaining primes can be split into n-cardinality halves with products both less than it.

The percentages by which a(n) exceeds the (3n)-th primorial's cube root are 60.9, 12.6, 3.38, 5.37, 1.02, 0.0883, 0.0340, 5.18*10^(-3), 5.01*10^(-4), 1.68*10^(-4), 2.37*10^(-5), 2.06*10^(-5), 3.87*10^(-5) and 1.14*10^(-6).

LINKS

James G. Merickel, Table of n, a(n) for n = 1..14

EXAMPLE

There are 15 distinct ways to break a set of six elements into three of cardinality 2 each. Among these for the first 6 primes, the partition {{2,13}, {3,11}, {5,7}} can be readily seen to give smallest possible maximum product of 35. So a(2)=35 (and with the convention that the product of a set consisting of one number is that number, a(1)=5 trivially).

PROG

(PARI)

{

p=vector(60, n, prime(n)); i=1; while(1,

  a=vectorsmall(3*i); for(j=1, i, a[j]=1);

  n=prod(j=1, i, p[j]); r=10^1000;

  P=prod(j=1, 3*i, p[j]); Q=P^(1/3);

  b=vectorsmall(3*i);

  for(j=2*i+1, 3*i, b[j]=1);

  while(1,

    if(n<r && n>Q,

        R=P/n; c=vector(2*i); k=1;

        for(j=1, 2*i, while(a[k], k++);

          c[j]=k; k++);

        d=vectorsmall(2*i); for(j=1, i, d[j]=1);

        e=vectorsmall(2*i); for(j=i+1, 2*i, e[j]=1);

        S=prod(j=1, i, p[c[j]]);

        while(1,

          if(S<n && R/S<n, r=n; break());

          if(d==e,

            break(),

            k=1; if(d[1]==0,

              while(d[k]==0, k++); z=1;

              while(d[k+1], S*=p[c[z]]/p[c[k]];

                d[z]=1; d[k]=0; z++; k++);

              d[k+1]=1; d[k]=0; S*=p[c[k+1]]/p[c[k]],

              while(d[k+1], k++); d[k]=0; d[k+1]=1;

              S*=p[c[k+1]]/p[c[k]]))));

  if(a==b,

    print1(r" "); break(),

    k=1; if(a[1]==0,

      while(a[k]==0, k++); z=1;

      while(a[k+1], n*=p[z]/p[k]; a[z]=1; a[k]=0; z++; k++);

      n*=p[k+1]/p[k]; a[k]=0; a[k+1]=1,

      while(a[k+1], k++); a[k]=0; a[k+1]=1; n*=p[k+1]/p[k]))); i++)

}

CROSSREFS

Cf. A260079.

Sequence in context: A194927 A089043 A317995 * A317816 A034236 A058015

Adjacent sequences:  A260072 A260073 A260074 * A260076 A260077 A260078

KEYWORD

nonn

AUTHOR

James G. Merickel, Jul 14 2015

STATUS

approved

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Last modified July 23 13:58 EDT 2019. Contains 325254 sequences. (Running on oeis4.)