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A206722 Parameters of Chebyshev function psi. 1
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 (list; graph; refs; listen; history; text; internal format)
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.

The first column is A000523; the second is A048766.

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-Fran├žois 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

Sequence in context: A096860 A128185 A175244 * A245222 A022300 A300983

Adjacent sequences:  A206719 A206720 A206721 * A206723 A206724 A206725

KEYWORD

nonn,tabf,easy

AUTHOR

John W. Nicholson, Feb 11 2012

STATUS

approved

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Last modified January 26 10:22 EST 2021. Contains 340436 sequences. (Running on oeis4.)