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A282870 a(n) = floor( Li(n) - pi(n) ). 2
0, 0, 0, 0, 1, 0, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 2, 1, 1, 2, 2, 1, 2, 2, 2, 3, 3, 2, 3, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 3, 3, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 3, 3, 4, 4, 3, 3, 3, 3, 3, 3, 3, 4, 3, 3, 3, 4, 3, 3, 3, 4, 4, 4, 3, 3, 4, 4, 4, 4, 5, 5, 4, 4, 4, 5, 4, 4, 3, 3, 4, 4 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,9
COMMENTS
Li(x) is the logarithmic integral of x.
pi(x) is the number of primes less than or equal to x, A000720(x).
"The Riemann hypothesis is an assertion about the size of the error term in the prime number theorem, namely, that pi(x) = li(x)+O(x^(1/2+epsilon))", see Nathanson, page 323.
REFERENCES
Melvyn B. Nathanson, Elementary Methods in Number Theory, Springer, 2000
LINKS
FORMULA
a(n) = A047783(n) - A000720(n).
MAPLE
a:= n-> floor(evalf(Li(n)))-numtheory[pi](n):
seq(a(n), n=2..120); # Alois P. Heinz, Feb 23 2017
MATHEMATICA
iend = 100;
For[x = 1, x <= iend, x++,
a[x] = N[LogIntegral[x] - PrimePi[x]]]; t =
Table[Floor[a[i]], {i, 2, iend}]; Print[t]
Table[Floor[LogIntegral[n] - PrimePi[n]], {n, 2, 110}] (* G. C. Greubel, May 17 2019 *)
PROG
(PARI) vector(110, n, n++; floor(real(-eint1(-log(n))) - primepi(n)) ) \\ G. C. Greubel, May 17 2019
(Magma) [Floor(LogIntegral(n) - #PrimesUpTo(n)): n in [2..110]]; // G. C. Greubel, May 17 2019
(Sage) [floor(li(n) - prime_pi(n)) for n in (2..110)] # G. C. Greubel, May 17 2019
CROSSREFS
Sequence in context: A026517 A072047 A327521 * A106802 A269254 A049236
KEYWORD
sign
AUTHOR
David S. Newman, Feb 23 2017
STATUS
approved

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Last modified July 25 02:50 EDT 2024. Contains 374585 sequences. (Running on oeis4.)