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

0, 7, 44, 299, 1940, 13549, 99987, 768752, 6089791, 49392723, 408550278, 3435528229, 29289695650, 252672394234, 2201981901415, 19360330918473, 171550299264139, 1530609037414453

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..18.

Siegfried "Zig" Herzog, Frequency of Occurrence of Prime Gaps

T. Oliveira e Silva, S. Herzog, and S. Pardi, Empirical verification of the even Goldbach conjecture and computation of prime gaps up to 4.10^18, Math. Comp., 83 (2014), 2033-2060.

Example

a(2) = 7 because there are 7 prime gaps of 6 below 10^2.

Mathematica

Accumulate@ Array[Count[Differences@ Prime@ Range[PrimePi[10^(# - 1) + 1], PrimePi[10^# - 1]], 6] &, 8] (* Michael De Vlieger, Apr 09 2021 *)

Prog

(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)

Crossrefs

Cf. A007508, A080841, A093737, A093739.

See also A023201, A031924.

Sequence in context: A099464 A355347 A254660 * A091127 A166775 A221541

Adjacent sequences: A093735 A093736 A093737 * A093739 A093740 A093741

Keyword

nonn,more

Author

Enoch Haga, Apr 15 2004.

Extensions

a(10)-a(13) from Washington Bomfim, Jun 22 2012

a(14)-a(18) from S. Herzog's website added by Giovanni Resta, Aug 14 2018

Status

approved