A263203 Consider the numbers m such that m = prime(k) + prime(k+2i+1) = prime(k+i) + prime(k+i+1) for some i and k. The sequence lists the number of pairs (i,k) giving the same value m = A105093(n).

1, 2, 2, 1, 4, 3, 4, 2, 2, 2, 1, 1, 2, 6, 5, 4, 1, 2, 1, 4, 4, 5, 7, 3, 6, 7, 1, 2, 1, 7, 10, 7, 7, 2, 6, 1, 5, 10, 12, 5, 10, 3, 5, 11, 9, 9, 8, 2, 6, 2, 2, 3, 10, 1, 5, 11, 10, 7, 7, 5, 3, 5, 5, 1, 4, 2, 4, 2, 5, 7, 4, 5, 8, 7, 6, 5, 3, 7, 13, 1, 1, 9, 5, 1

Offset: 1

Author

Michel Lagneau, Oct 12 2015

Keywords

• nonn

Comments

The form m = prime(k) + prime(k+2i+1) = prime(k+i) + prime(k+i+1) is a generalization of A105093 (form prime(k) + prime(k+3) = prime(k+1) + prime(k+2)), and the set of the numbers m is exactly A105093(n).

Examples

			a(6) = 3 because A105093(6)= 84 and:
for (i,k)=(1,12), prime(12)+ prime(15)= prime(13)+ prime(14)=37+47=41+43=84;
for (i,k)=(2,11), prime(11)+ prime(16)= prime(12)+ prime(15)=31+53=37+47=84;
for (i,k)=(4,9), prime(9)+ prime(18)= prime(13)+ prime(14)=23+61=41+43=84.
So, we find 3 pairs (i,k) giving m = 84.
		

Crossrefs

Cf. A105093.

Programs

Maple

with(numtheory):nn:=5000:
A105093:={18,24,30,36,60,84,120,162,204,210,216,240,288,330,372,390,456,520,540,624,726,762,798,840,852,882,924,978,990,1104,1140,1164,1200,1392,1410,1428,1530,1632,1650,1716,1740,1764,1794,1848,1974,2052,2100,2112,2184,2226,2334,2460,2580,2604,2688,2856,2970,2976,3054,3102,3138,3150,3180,3240,3500,3536,3612,3744,3750,3882,3966,3996,4056,4092,4170,4242,4680,4698,4728,4782,4810,5100,5376,5460}:n0:=nops(A105093):
  for n from 1 to n0 do:
   ii:=0:it:=0:q:=A105093[n]:
   for i from 1 to 100 do:
      for k from 1 to nn do:
        s1:=ithprime(k)+ithprime(k+2*i+1):
        s2:= ithprime(k+i)+ithprime(k+i+1):
        if s1=s2 and s1=q
        then
        it:=it+1:
        else
        fi:
       od:
      od:
       printf(`%d, `,it):
     od: