A290507 Sums and differences of products of the first n primes partitioned into two disjoint parts.

1, 5, 7, 11, 13, 17, 23, 29, 31, 37, 41, 47, 67, 73, 83, 89, 97, 101, 103, 107, 127, 131, 139, 151, 157, 163, 169, 179, 181, 199, 221, 227, 239, 241, 263, 307, 313, 323, 337, 347, 349, 353, 359, 361, 379, 383, 389, 391, 397, 421, 457, 463, 467, 491, 499, 521, 527, 601, 619, 643, 653, 667, 673, 709, 713

Offset

1,2

Comments

Partition the set Pn = {2,3,5,...,pn} of the first n primes into two disjoint parts. Let a,b be their respective products, S = a + b and D = |a - b|. The list is composed of sorted values of S and D.

Numbers a,b share no common factors. It follows that the prime factors of S or D can't divide either a or b. So the smallest possible prime factor of S or D is pn+1.

After the value 1, the next 25 numbers of the list are primes. Then the proportion of primes decreases. For the first 2000 elements, about 50% are primes.

Links

Table of n, a(n) for n=1..65.

C. Aebi and G. Cairns, Partitions of Primes, Parabola, Vol. 45, No. 1 (2009).

Example

3 - 2 = 1
2 + 3 = 5
2*5 - 3 = 7
2*3 + 5 = 11
2*5 + 3 = 13
3*5 + 2 = 17
2*3*5 - 7 = 23
2*7 + 3*5 = 29
2*5 + 3*7 = 31
...

Prog

(JavaScript)
<script>
/**************************
Sum and Difference of multiplication of two complementary subset the n first primes
****************************/
// List of the first primes
var P=new Array(2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97);
function Sum_and_Difference(n)
{
  var L=[];  // Table used to have the two partitions of a set of elements
  var a=1;  // First product of primes
  var b=1;  // Second product of primes
  var Ok = 0; // if n is only composed of 1
  var S, D;  // Sum and Difference of products of primes
  // Generate a table L containing n in binary digits : 0 and 1 values in L are used to obtain the two partitions
  do
  {
var r=n%2;
L.push(r);
n=(n-r)/2;
  }while(n!=0);
  for(var i=0; i < L.length ; i++)
  {
     if(L[i] == 0)
     {
       Ok = 1;
       a = a * P[i];
     }
     else
       b = b * P[i];
  }
  if (Ok)
  {
    D = Math.abs(a-b);
    document.write(D + " =  |" + a + " -  " + b + "| </BR> ");
    S = a+b;
    document.write(S + " = " + a + " +  " + b + " </BR> ");
  }
}
for (var j = 1 ; j < 32 ; j++)
{
Sum_and_Difference(j);
}
</script>

Crossrefs

Sequence in context: A104860 A216751 A216741 * A139812 A180951 A040146

Adjacent sequences: A290504 A290505 A290506 * A290508 A290509 A290510

Keyword

nonn

Author

Yves Debeuret, Aug 04 2017

Status

approved