A093744 - OEIS

%I #20 May 11 2023 11:57:29

%S 0,0,1,40,514,4909,43851,384738,3351032,29189691,255371697,2246576317,

%T 19878698732,176913444597,1583135619012,14240370591853,

%U 128711774414592,1168583066366245

%N Number of prime pairs below 10^n having a difference of 18.

%H Siegfried "Zig" Herzog, <a href="http://zigherzog.net/primes/index.html#compare">Frequency of Occurrence of Prime Gaps</a>

%H T. Oliveira e Silva, S. Herzog, and S. Pardi, <a href="http://dx.doi.org/10.1090/S0025-5718-2013-02787-1">Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18</a>, Math. Comp., 83 (2014), 2033-2060.

%e a(4) = 40 because there are 40 prime gaps of 18 below 10^4.

%t Table[Count[Differences[Prime[Range[PrimePi[10^n]]]],18],{n,8}] (* The program generates the first 8 terms in the sequence. To generate more increase the n constant but the program may take a long time to run. *) (* _Harvey P. Dale_, Sep 03 2021 *)

%o (UBASIC) 20 N=1:dim T(34); 30 A=nxtprm(N); 40 N=A; 50 B=nxtprm(N); 60 D=B-A; 70 for x=2 to 34 step 2; 80 if D=X and B<10^2+1 then T(X)=T(X)+1; 90 next X; 100 if B>10^2+1 then 140; 110 B=A; 120 N=N+1; 130 goto 30; 140 for x=2 to 34 step 2; 150 print T(X);, 160 next (This program simultaneously finds values from 2 to 34 - if gap=2 add 1- adjust lines 80 and 100 for desired 10^n)

%Y Cf. A007508, A093743, A093745.

%K nonn,more

%O 1,4

%A _Enoch Haga_, Apr 15 2004

%E a(10)-a(13) from _Washington Bomfim_, Jun 22 2012

%E a(14)-a(18) from S. Herzog's website added by _Giovanni Resta_, Aug 14 2018