A206722 Parameters of Chebyshev function psi.

1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 3, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1, 4, 2, 1, 1, 1, 1, 1

Offset: 2

Author

John W. Nicholson, Feb 11 2012

Keywords

• nonn
• tabf
• easy

Comments

a(x,n) is the exponent k such that prime(n)^k <= x and x < prime(n)^(k+1).

psi(x) = Sum_{p_n <= x} k*log(p_n), where a(x,n) = k is the unique integer such that p_n^k <= x but p_n^(k+1) > x.

Related to Firoozbakht's Conjecture (1982): p_n^(1/n) > p_(n+1)^(1/(n+1)) for all n >= 1.

Examples

			If x = 7, then 2^2, 3^1, 5^1, 7^1 <= x < 2^3, 3^2, 5^2, 7^2, respectively so k = 2, 1, 1, 1, respectively.
The table starts in row x=2 with columns n >= 1 as:
  1;
  1, 1;
  2, 1;
  2, 1, 1;
  2, 1, 1;
  2, 1, 1, 1;
  3, 1, 1, 1;
  3, 2, 1, 1;
  3, 2, 1, 1, 1;
		

Links

• N. Kanti Sinha, On a new property of primes that leads to a generalization of Cramer's conjecture, arXiv:1010.1399 [math.NT], 2010.
• Wikipedia, Chebyshev function

Crossrefs

Columns: A000523 (n=1), A062153 (n=2).

Programs

Mathematica

A206722[x_, n_] := Module[{p = Prime[n]}, For[k = 0, True, k++, If[p^(k+1) > x && p^k <= x, Return[k]]]];
Table[DeleteCases[Table[A206722[x, n], {n, 1, 17}], 0], {x, 2, 20}] // Flatten (* Jean-François Alcover, Sep 15 2018, after R. J. Mathar *)

Maxima

prime(n) := block(
    if n = 1 then
       return(2)
    else
    return(next_prime(prime(n-1)))
)$ /* very slow recursive definition of A000040 */
A206722(x,n) := block(
    local(p),
    p : prime ( n ),
    for k : 0 do (
       if p^(k+1) > x and p^k <= x then
          return(k)
       )
)$
for x : 2 thru 20 do (
    for n : 1 thru 17 do
      sprint(A206722(x,n)),
    newline()
)$ /* R. J. Mathar, Feb 14 2012 */