A205601 Goldbach's problem extended to division: number of decompositions of 2n into the floor of unordered ratios of two primes, floor(q/p) = 2n, where p < 2n < q.

0, 1, 3, 5, 4, 5, 10, 5, 10, 16, 12, 17, 18, 16, 19, 27, 23, 22, 34, 27, 34, 39, 39, 45, 51, 41, 50, 51, 44, 57, 68, 71, 63, 74, 63, 76, 87, 84, 89, 104, 94, 108, 111, 99, 117, 116, 120, 104, 126, 114, 133, 146, 149, 146, 166, 148, 190, 178, 182, 170, 179, 173

Offset

1,3

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Example

For n = 3, a(n) = 3 because 6 is the floor of 13/2, 19/3, and 31/5. - T. D. Noe, Jan 31 2012

Mathematica

Table[Length[Flatten[Table[Select[2*n*p + Range[p - 1], PrimeQ], {p, Prime[Range[PrimePi[2*n - 1]]]}]]], {n, 62}] (* T. D. Noe, Jan 31 2012 *)

Prog

(C++) #include <iostream>
using namespace std;
int main() //C++ code for the first 20 even integers >= 2
           //where floor(q/p) = 2n, p<2n<q, by James D. Klein
{ bool a[2000];  //Initialize array p with primes
  int p[304];
  int n=2000, i=0;
  a[0] = a[1] = false;
  for(int j=2; j<n; j++) a[j] = true;
  for(int j=2; j<=n/2; j++)
    for(int k=2; k<=n/j; k++)
      a[j*k] = false;
  for(int j=2; j<n; j++)
    if(a[j]) p[i++]=j;
  int count, istart = 0; //Generate the decompositions
  for(int n=1; n<=20; n++)
  { while(2*n>p[istart]) istart++;
    count = 0;
    for(int j=0; p[j]<2*n; j++)
      for(int i=istart; p[i]<(p[j]+1)*2*n; i++)
        if(p[i]/p[j]==2*n)count++;
      cout << n << ". " << count << endl;
  }
return 0;
}
(PARI) a(n)=n*=2; my(s, t); forprime(p=2, n-1, t=n*p; while(n==(t=nextprime(t+1))\p, s++)); s \\ Charles R Greathouse IV, Jan 30 2012

Crossrefs

Cf. A002375, A202472.

Sequence in context: A206035 A200100 A057759 * A021286 A200324 A063259

Adjacent sequences: A205598 A205599 A205600 * A205602 A205603 A205604

Keyword

nonn

Author

James D. Klein, Jan 29 2012

Extensions

a(21)-a(62) from Charles R Greathouse IV, Jan 31 2012

Status

approved