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A206722 Parameters of Chebyshev function psi.

%I A206722 #38 Feb 18 2024 01:26:08
%S A206722 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,
%T A206722 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,
%U A206722 4,2,1,1,1,1,1
%N A206722 Parameters of Chebyshev function psi.
%C A206722 a(x,n) is the exponent k such that prime(n)^k <= x and x < prime(n)^(k+1).
%C A206722 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.
%C A206722 Related to Firoozbakht's Conjecture (1982): p_n^(1/n) > p_(n+1)^(1/(n+1)) for all n >= 1.
%H A206722 N. Kanti Sinha, <a href="https://arxiv.org/abs/1010.1399">On a new property of primes that leads to a generalization of Cramer's conjecture</a>, arXiv:1010.1399 [math.NT], 2010.
%H A206722 Wikipedia, <a href="http://en.wikipedia.org/wiki/Chebyshev_function#Relationships">Chebyshev function</a>
%e A206722 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.
%e A206722 The table starts in row x=2 with columns n >= 1 as:
%e A206722   1;
%e A206722   1, 1;
%e A206722   2, 1;
%e A206722   2, 1, 1;
%e A206722   2, 1, 1;
%e A206722   2, 1, 1, 1;
%e A206722   3, 1, 1, 1;
%e A206722   3, 2, 1, 1;
%e A206722   3, 2, 1, 1, 1;
%t A206722 A206722[x_, n_] := Module[{p = Prime[n]}, For[k = 0, True, k++, If[p^(k+1) > x && p^k <= x, Return[k]]]];
%t A206722 Table[DeleteCases[Table[A206722[x, n], {n, 1, 17}], 0], {x, 2, 20}] // Flatten (* _Jean-François Alcover_, Sep 15 2018, after _R. J. Mathar_ *)
%o A206722 (Maxima)
%o A206722 prime(n) := block(
%o A206722     if n = 1 then
%o A206722        return(2)
%o A206722     else
%o A206722     return(next_prime(prime(n-1)))
%o A206722 )$ /* very slow recursive definition of A000040 */
%o A206722 A206722(x,n) := block(
%o A206722     local(p),
%o A206722     p : prime ( n ),
%o A206722     for k : 0 do (
%o A206722        if p^(k+1) > x and p^k <= x then
%o A206722           return(k)
%o A206722        )
%o A206722 )$
%o A206722 for x : 2 thru 20 do (
%o A206722     for n : 1 thru 17 do
%o A206722       sprint(A206722(x,n)),
%o A206722     newline()
%o A206722 )$ /* _R. J. Mathar_, Feb 14 2012 */
%Y A206722 Columns: A000523 (n=1), A062153 (n=2).
%K A206722 nonn,tabf,easy
%O A206722 2,4
%A A206722 _John W. Nicholson_, Feb 11 2012