cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A139677 Estimate of the sum of twin prime pairs < 10^n = 4*Pi2(10^2n).

Original entry on oeis.org

32, 820, 24676, 1761248, 109650716, 7482340880, 543121286660, 41216742789192
Offset: 1

Views

Author

Cino Hilliard, Jun 13 2008

Keywords

Comments

Since we have SumTP(n) up to n=10^12, we can reverse this process and estimate Pi2(n) for n = 18,20,22,24. Since 4*Pi2(2n) ~ SumTP(n), Pi2(2n) ~ SumTP(n)/4.
The link shows these estimates and the relative error. Also estimated is the odd values 17,19,21,23,25 by curve fitting 6 points to a 5th degree polynomial to the base-10 log of the values and interpolating.

Examples

			For n = 8, SumTP(8) = A118552(8) = 41205774636932. Pi2(16)= 10304185697298.
4*Pi2(16) = 41216742789192. This has an error of 0.00026...
		

Crossrefs

Formula

Pi2(n) is the twin prime counting function = number of twin prime pairs < n. a(n) = 4*A007508(2n) for n <= 8. SumTP(n) = sum of twin prime pairs < n.

A146536 Sum of the first 10^n twin primes.

Original entry on oeis.org

8, 908, 328184, 69004076, 11556327260, 1707243198956, 237064232862404, 31153163750203064, 3947120494191630260, 486665774050923191336, 58727077924563028184984
Offset: 0

Views

Author

Cino Hilliard, Oct 31 2008

Keywords

Comments

The author's Gcc/Gmp program is in the links section. The page also has the PARI bisection algorithms which give a very good approximation for the n-th prime number and the n-th lower twin prime number. The first 5 terms can be computed from the PARI script although 11556327260 takes 4 hours on a 2.53ghz 2 gig ram p4.

Examples

			The 10^0-th twin prime pair is (3,5). This adds up to 8, the first entry in the sequence.
The first 10^1 twin prime pairs are (3,5),(5,7),(11,13) (17,19),(29,31),(41,43),(59,61),(71,73),(101,103),(107,109). This adds up to 908, the second entry in the table.
		

Crossrefs

Programs

  • C
    // See links.
    
  • PARI
    a(n)={my(k=10^n,s=k,q); forprime(p=3, oo, if(p==q+2, s+=q; k--; if(!k, return(2*s))); q=p)} \\ Andrew Howroyd, Oct 22 2023

Formula

a(n) = A086169(10^n). - Andrew Howroyd, Oct 22 2023

Extensions

a(10) corrected by Bill McEachen, Oct 16 2023

A129697 Sum of isolated primes < 10^n.

Original entry on oeis.org

2, 577, 51896, 4009989, 345281974, 30126035380, 2657646397769, 238004015750349, 21505022353019864, 1960179022139638131, 180020101551309284879, 16639947666244921992434, 1546703841471331792075283, 144470014127904647017055666
Offset: 1

Views

Author

Cino Hilliard, Jun 08 2007

Keywords

Comments

Isolated primes are primes that are not twin prime members.

Examples

			The sum of the isolated primes < 100 = 2+23+37+47+53+67+79+83+89+97 = 577, the second term of this sequence.
		

Crossrefs

Programs

  • PARI
    isoprimes(n) = { local(j,s,x); for(j=1,n, s=0; forprime(x=2,10^j, if(!isprime(x-2)&&!isprime(x+2),s+=x) ); print1(s", ") ) }

Formula

a(n) = A046731(n) - A118552(n) + 5.

Extensions

a(13)-a(14) from Amiram Eldar, Jul 02 2024
Showing 1-3 of 3 results.