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A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.

%I A093738 #39 Jul 09 2025 04:23:24
%S A093738 0,7,44,299,1940,13549,99987,768752,6089791,49392723,408550278,
%T A093738 3435528229,29289695650,252672394234,2201981901415,19360330918473,
%U A093738 171550299264139,1530609037414453
%N A093738 Number of pairs of consecutive prime (p,q) with q-p=6 and q < 10^n.
%C A093738 Note that one has to be careful to distinguish between pairs of consecutive primes (p,q) with q-p = 6 (A031924), and pairs of primes (p,q) with q-p = 6 (A023201). Here we consider the former, whereas A080841 considers the latter. - _N. J. A. Sloane_, Mar 07 2021
%H A093738 Siegfried "Zig" Herzog, <a href="http://zigherzog.net/primes/index.html#compare">Frequency of Occurrence of Prime Gaps</a>
%H A093738 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 A093738 a(2) = 7 because there are 7 prime gaps of 6 below 10^2.
%t A093738 Accumulate@ Array[Count[Differences@ Prime@ Range[PrimePi[10^(# - 1) + 1], PrimePi[10^# - 1]], 6] &, 8] (* _Michael De Vlieger_, Apr 09 2021 *)
%o A093738 (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 A093738 Cf. A007508, A080841, A093737, A093739.
%Y A093738 See also A023201, A031924.
%K A093738 nonn,more
%O A093738 1,2
%A A093738 _Enoch Haga_, Apr 15 2004
%E A093738 a(10)-a(13) from _Washington Bomfim_, Jun 22 2012
%E A093738 a(14)-a(18) from S. Herzog's website added by _Giovanni Resta_, Aug 14 2018