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

2,4

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.

Links

Table of n, a(n) for n=2..76.

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

Example

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;

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 *)

Prog

(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 */

Crossrefs

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

Sequence in context: A096860 A128185 A175244 * A245222 A022300 A347552

Adjacent sequences: A206719 A206720 A206721 * A206723 A206724 A206725

Keyword

nonn,tabf,easy

Author

John W. Nicholson, Feb 11 2012

Status

approved