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
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
KEYWORD
nonn,tabf,easy
AUTHOR
John W. Nicholson, Feb 11 2012
STATUS
approved