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A038625 a(n) is smallest number m such that m = n*pi(m), where pi(k) = number of primes <= k (A000720).

%I A038625 #54 Feb 16 2025 08:32:38
%S A038625 2,27,96,330,1008,3059,8408,23526,64540,175197,480852,1304498,3523884,
%T A038625 9557955,25874752,70115412,189961182,514272411,1394193580,3779849598,
%U A038625 10246935644,27788566029,75370121160,204475052375,554805820452,1505578023621,4086199301996,11091501630949
%N A038625 a(n) is smallest number m such that m = n*pi(m), where pi(k) = number of primes <= k (A000720).
%C A038625 Golomb shows that solutions exist for each n>1.
%C A038625 Equivalently, for n > 1, least m such that m >= n*pi(m). - _Eric M. Schmidt_, Aug 05 2014
%C A038625 The values a(26),...,a(50) were calculated with the Eratosthenes sieve making use of strong bounds for pi(x), which follow from partial knowledge of the Riemann hypothesis, and the analytic method for calculating initial values of pi(x). - _Jan Büthe_, Jan 16 2015
%H A038625 Jan Büthe, <a href="/A038625/b038625.txt">Table of n, a(n) for n = 2..50</a>
%H A038625 John D. Cook, <a href="https://www.johndcook.com/blog/2018/10/17/integer-odds-and-prime-numbers/">Integer odds and prime numbers</a>
%H A038625 S. W. Golomb, <a href="http://www.jstor.org/stable/2312732">On the Ratio of N to pi(N)</a>, American Mathematical Monthly, 69 (1962), 36-37.
%H A038625 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeCountingFunction.html">Prime Counting Function.</a>
%F A038625 It appears that a(n) is asymptotic to e^2*exp(n). - _Chris K. Caldwell_, Apr 02 2008
%F A038625 a(n) = A038626(n) * n. - _Max Alekseyev_, Oct 13 2023
%e A038625 pi(3059) = 437 and 3059/437 = 7, so a(7)=3059.
%p A038625 with(numtheory); f:=proc(n) local i; for i from 2 to 10000 do if i mod pi(i) = 0 and i/pi(i) = n then RETURN(i); fi; od: RETURN(-1); end; # _N. J. A. Sloane_, Sep 01 2008
%t A038625 t = {}; k = 2; Do[While[n*PrimePi[k] != k, k++]; AppendTo[t, k], {n, 2, 15}]; t (* _Jayanta Basu_, Jul 10 2013 *)
%o A038625 (PARI)
%o A038625 a(n)=my(k=1); while(k!=n*primepi(k),k++); k;
%o A038625 for (n=2, 20, print1(a(n), ", ")); \\ _Derek Orr_, Aug 13 2014
%o A038625 (Python)
%o A038625 from math import exp
%o A038625 from sympy import primepi
%o A038625 def a(n):
%o A038625   m = 2 if n == 2 else int(exp(n)) # pi(m) > m/log(m) for m >= 17
%o A038625   while m != n*primepi(m): m += 1
%o A038625   return m
%o A038625 print([a(n) for n in range(2, 10)]) # _Michael S. Branicky_, Feb 27 2021
%Y A038625 Cf. A000720, A073436, A038623, A038624, A038625, A038626, A038627, A057809.
%K A038625 nonn
%O A038625 2,1
%A A038625 _Jud McCranie_
%E A038625 Three more terms from _Labos Elemer_, Sep 12 2003
%E A038625 Edited by _N. J. A. Sloane_ at the suggestion of _Chris K. Caldwell_, Apr 08 2008
%E A038625 24 terms added and entry a(26) corrected by _Jan Büthe_, Jan 07 2015