• Journal of applied mathematics & informatics
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• v.22 no.1_2
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• pp.441-452
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• 2006

We introduce the counting function ${\pi}^*_{2.8}(x)$ of the primes with difference 8 between consecutive primes ($p_n,\;p_{n+l}=p_n+8$) can be approximated by logarithm integral $Li^*_{2.8}$. We calculate the values of ${\pi}^*_{2.8}(x)$ and the sum $C_{2,8}(x)$ of reciprocals of primes with difference 8 between consecutive primes $p_n,\;p_{n+l}=p_n+8$ where x is counted up to $7{\times}10^{10}$. From the results of these calculations. we obtain ${\pi}^*_{2.8}(7{\times}10^{10}$)= 133295081 and $C_{2.8}(7{\times}10^{10}) = 0.3374{\pm}2.6{\times}10^{-4}$.

• Journal of applied mathematics & informatics
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• v.13 no.1_2
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• pp.37-51
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• 2003

Enumeration of the primes with difference 4 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function ,r2,4(5${\times}$10$\^$10/) = l18905303. The sum of reciprocals of primes with gap 4 between consecutive primes is computed B$_4$(5 ${\times}$ 10$\^$10/) = 1.1970s4473029 and B$_4$ = 1.197054 ${\pm}$ 7 ${\times}$ 10$\^$-6/. And Enumeration of the primes with difference 6 between consecutive primes, is counted up to 5${\times}$10$\^$10/, yielding the counting function $\pi$$\_$2.6/(5${\times}$10$\^$10/) = 215868063. The sum of reciprocals of primes with gap 6 between consecutive primes is computed B$\_$6/(5${\times}$10$\^$10/) = 0.93087506039231 and B$\_$6/ = 1.135835 ${\pm}$ 1.2${\times}$10$\^$-6/.

• Journal of Internet of Things and Convergence
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• v.6 no.4
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• pp.49-57
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• 2020

In this paper, I provided an approximation function Li*2,10(x) using logarithm integral for the counting function π*2,10(x) of consecutive deca primes. Several personal computers and Mathematica were used to validate the approximation function Li*2,10(x). I found the real value of π*2,10(x) and approximate value of Li*2,10(x) for various x ≤ 1011. By the result of theses calculations, most of the error rates are margins of error of 0.005%. Also, I proved that the sum C2,10(∞) of reciprocals of all primes with difference 10 between primes is finite. To find C2,10(∞), I computed the sum C2,10(x) of reciprocals of all consecutive deca primes for various x ≤ 1011 and I estimate that C2,10(∞) probably lies in the range C2,10(∞)=0.4176±2.1×10-3.