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

1,1

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.

Links

Table of n, a(n) for n=1..24.

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

Formula

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

Example

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

Crossrefs

Cf. A006988, A074383.

Sequence in context: A264028 A078741 A248461 * A331056 A274877 A091014

Adjacent sequences: A129867 A129868 A129869 * A129871 A129872 A129873

Keyword

nonn,more

Author

Cino Hilliard, Jun 04 2007

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

Status

approved