[seqfan] An interesting sequence connected with primes

[Vladimir Shevelev]

Dear SeqFans,
 
I would like to discuss an astonishing behavior of the following sequence connected with primes ( I submitted it as A229512):
a(n) is the minimal k, such that nextprime(2*k+1)-2*k=prime(n). It begins
 
0,1,3,11,58,57,262,261,564,666,665,4775,7843,7842,9807,9804,15705,15704,15701,15699,15698,
77964,77962,180330,180326,185136,185135,185133,185132,185130,678603,678601,1005372,
1005371,1005366,2326178,8525865,8525862,...
 
It is interesting the distribution of terms over gropes of close magnitudes. For example, 58,57; 9807,9804;, 15704,15701,15699,15698; etc. Could anyone to give an explanation? 
 
Best regards,
Vladimir

 Shevelev Vladimir‎