1,3

I introduce this sequence which is A128913(n)/2 because it is closely related

to the prime counting function Pi(n) and the sum of primes < n for large n.

This is, SumP(n) ~ n*Pi(n)/2. For n = 10^10 n*Pi(n)/2 = 2275262555000000000.

Sum primes < 10^n = 2220822432581729238. This has error 0.0245...For the

largest known sum of primes, for sums < 10^20, we have n*Pi(n)/2 =

111040980128045942000000000000000000000. The sum of primes < 10^20 =

109778913483063648128485839045703833541. The error here is -0.01149... It

converges quite slowly and better approximations have been found.

This relationship was derived by using the summation formula for an arithmetic

progression. For the odd integers where n is even, let the first term = 1, the

last term is n-1 and the number of terms is n/2. So the sum of the odd numbers

< n is ((1 +n-1)*n/2)/2. If we let Pi(x) be the number of terms, we get the

result n*Pi(n)/2. A closed formula, SumP(n) ~ n^2/(2*log(n)-1) is quite

accurate. The best formula I have found is the remarkable SumP(n) ~ Pi(n^2).

This formula has an error of 6.162071097138 E-11 for the largest known sum of

primes or sum < 10^20.

See the link Sum of Primes for derivations of these asymptotic formulas.

Cino Hilliard, Sum of Primes.

Pi(n) is the prime counting function, the number of primes < n. Define SumP(n) is the sum of primes < n.

(PARI) g(n) = for(x=1, n, print1(floor(x*primepi(x)/2)", "))

Cf. A128913.

Sequence in context: A061568 A146994 A103054 this_sequence A098390 A008763 A005896

Adjacent sequences: A140205 A140206 A140207 this_sequence A140209 A140210 A140211

nonn,uned

Cino Hilliard (hillcino368(AT)hotmail.com), Jun 09 2008