A263203 - OEIS

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

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OFFSET

1,2

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).

LINKS

Table of n, a(n) for n=1..84.

EXAMPLE

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.

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: