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

32, 820, 24676, 1761248, 109650716, 7482340880, 543121286660, 41216742789192

Offset: 1

Author

Cino Hilliard, Jun 13 2008

Keywords

• nonn
• more

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...
		

Links

• Cino Hilliard, SumPrimes. [Broken link]
• Bill McEachen, Reconstructed Hilliard file

Crossrefs

Cf. A118552, A007508.

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.