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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. 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 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

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Last modified January 24 00:05 EST 2022. Contains 350515 sequences. (Running on oeis4.)