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

%I A290507 #17 Jun 23 2022 20:37:10
%S A290507 1,5,7,11,13,17,23,29,31,37,41,47,67,73,83,89,97,101,103,107,127,131,
%T A290507 139,151,157,163,169,179,181,199,221,227,239,241,263,307,313,323,337,
%U A290507 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
%N A290507 Sums and differences of products of the first n primes partitioned into two disjoint parts.
%C A290507 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.
%C A290507 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.
%C A290507 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.
%H A290507 C. Aebi and G. Cairns, <a href="https://www.parabola.unsw.edu.au/2000-2009/volume-45-2009/issue-1/article/partitions-primes">Partitions of Primes</a>, Parabola, Vol. 45, No. 1 (2009).
%e A290507 3 - 2 = 1
%e A290507 2 + 3 = 5
%e A290507 2*5 - 3 = 7
%e A290507 2*3 + 5 = 11
%e A290507 2*5 + 3 = 13
%e A290507 3*5 + 2 = 17
%e A290507 2*3*5 - 7 = 23
%e A290507 2*7 + 3*5 = 29
%e A290507 2*5 + 3*7 = 31
%e A290507 ...
%o A290507 (JavaScript)
%o A290507 <script>
%o A290507 /**************************
%o A290507 Sum and Difference of multiplication of two complementary subset the n first primes
%o A290507 ****************************/
%o A290507 // List of the first primes
%o A290507 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);
%o A290507 function Sum_and_Difference(n)
%o A290507 {
%o A290507   var L=[];  // Table used to have the two partitions of a set of elements
%o A290507   var a=1;  // First product of primes
%o A290507   var b=1;  // Second product of primes
%o A290507   var Ok = 0; // if n is only composed of 1
%o A290507   var S,D;  // Sum and Difference of products of primes
%o A290507   // Generate a table L containing n in binary digits : 0 and 1 values in L are used to obtain the two partitions
%o A290507   do
%o A290507   {
%o A290507 var r=n%2;
%o A290507 L.push(r);
%o A290507 n=(n-r)/2;
%o A290507   }while(n!=0);
%o A290507   for(var i=0;i < L.length ; i++)
%o A290507   {
%o A290507      if(L[i] == 0)
%o A290507      {
%o A290507        Ok = 1;
%o A290507        a = a * P[i];
%o A290507      }
%o A290507      else
%o A290507        b = b * P[i];
%o A290507   }
%o A290507   if (Ok)
%o A290507   {
%o A290507     D = Math.abs(a-b);
%o A290507     document.write(D + " =  |" + a + " -  " + b + "| </BR> ");
%o A290507     S = a+b;
%o A290507     document.write(S + " = " + a + " +  " + b + " </BR> ");
%o A290507   }
%o A290507 }
%o A290507 for (var j = 1 ; j < 32 ; j++)
%o A290507 {
%o A290507 Sum_and_Difference(j);
%o A290507 }
%o A290507 </script>
%K A290507 nonn
%O A290507 1,2
%A A290507 _Yves Debeuret_, Aug 04 2017