A129870 Difference between the (10^n)-th and the (10^n-1)-th prime.

6, 18, 12, 6, 20, 6, 2, 4, 12, 12, 20, 12, 22, 26, 30, 6, 72, 152, 72, 24, 30, 96, 124, 50

Offset: 1

Author

Cino Hilliard, Jun 04 2007

Keywords

• nonn
• more

Comments

It is interesting that the number 2 occurs deep into the sequence indicating a twin prime pair. It is reasonable to ask if this will ever occur again. Similarly, the analogous sequence A074383, "Difference between (1+10^n)-th and (10^n)-th primes" has 2 occurring shallow into the sequence. It is reasonable to ask if the number 2 will ever occur again in that sequence. The link provides an excellent algorithm, primex(n), that I developed to find the n-th prime using Gram's approximation of Riemann's approximation R(x) for Pi(x). Primex(n) will give about n/2 exact digits for prime(n). For A006988 (18), primex(18) is 44211790234127235469.62904554...This is only as good as R (x) but nevertheless is superior to the exact formulas out there from a practical stand point. If we apply the code gpx(n) = for(x=1,n,y=nextprime(primex(10^x))-nextprime (primex(10^x-1));print1(floor(y)",")), we will get the erratic concoction 2,0,8,14,22,28,26,0,72,18,22,0,0,0,0,0,32,0,80,78,60,0 as an analytical counterpart of the sequence given.

Examples

			The (10^18)-th prime or A006988(18) = 44211790234832169331.
Using PARI, precprime(A006988(18)-1) = 44211790234832169179.
The difference is a(18) = 152.
		

Links

• C. Hilliard, Nth prime approx [broken link].

Crossrefs

Cf. A006988, A074383.

Formula

a(n) = A006988(n)-A151799(A006988(n))

Extensions

a(19) from Max Alekseyev, May 13 2009

a(20) from Max Alekseyev, May 30 2013

a(21),a(22) from Max Alekseyev, Dec 04 2014

a(23)-a(24) from Chai Wah Wu using terms in A006988, Sep 18 2018