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A206722
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Parameters of Chebyshev function psi.
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1
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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
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OFFSET
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2,4
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COMMENTS
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a(x,n) is the exponent k such that prime(n)^k <= x and x < prime(n)^(k+1).
psi(x) = sum(with 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 the Firoozbakht’s Conjecture (1982): p_n^(1/n) > p_(n+1)^(1/(n+1)) for all n ≥ 1.
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LINKS
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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,/a>, arXiv:1010.1399
Wikipedia, Chebyshev function
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EXAMPLE
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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;
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PROG
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(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 */
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CROSSREFS
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Sequence in context: A096860 A128185 A175244 * A022300 A105690 A214364
Adjacent sequences: A206719 A206720 A206721 * A206723 A206724 A206725
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KEYWORD
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nonn,tabf,easy
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AUTHOR
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John W. Nicholson, Feb 11 2012
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STATUS
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approved
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