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%I #32 Aug 25 2019 13:31:45
%S 4,498,1139556,33182655688
%N Greatest number N such that the number of primes (<= N) <= the number of base-n-zerofree numbers (<= N).
%C Further terms:
%C 14670238462896430 < a(6) < 14670469667698570;
%C 1.88655928870547380*10^22 < a(7) < 1.8865698003644555*10^22;
%C 1.5845871199286*10^29 < a(8) < 1.5845909238805*10^29;
%C 3.7023896360635*10^36 < a(9) < 3.7023941021398*10^36;
%C 1.0075615552422*10^45 < a(10) < 1.0075622026833*10^45;
%C 1.3480809599483*10^53 < a(11) < 1.3480814844466*10^53;
%C 3.9618565460983*10^62 < a(12) < 3.9618574860993*10^62;
%C 7.8648507615953*10^71 < a(13) < 7.864851991241*10^71;
%C 4.7945106758325*10^81 < a(14) < 4.7945111864185*10^81;
%C 1.0953005932169*10^92 < a(15) < 1.0953006746693*10^92;
%C 8.3149001821943*10^148 < a(20) < 8.3149003278317*10^148.
%C All terms are even, since a(n) + 1 is always an odd prime number.
%C The numbers a(n) + 1 and a(n) + 2 are zero containing numbers (in base n).
%C The numbers between a(n) and p := min( k > a(n) + 1 | k is prime) are zero containing numbers, i.e., a(n) + j is a zero containing number for 0 < j < p - a(n).
%C For numbers m > a(10) = 1.00756...*10^45, we have pi(m) > A324161(m) [= number of zerofree numbers <= m]; in general, the ratio A324161(m) to pi(m) is O(log(n)*n^d), where d := 1 - 1/(1 - log_10(9)) = -0.0457..., and thus tends to 0 for m --> infinity. Consequently, the share of primes <= m which have no '0'-digit become significantly smaller as m rises beyond that bound a(10). For m = 10^100, the share is not greater than 0.000688, for m = 10^1000, the share cannot exceed 4.52757*10^(-43). Conversely, the share of primes which contain a '0'-digit tends to 1 as m grows to infinity (cf. A011540).
%C Conjecture: a(n) can be represented in the form a(n) = k*n^m + j, where j < (n^(m+1)-1)/(n-1) - n^m, and m > 1, 0 < k < n.
%F a(n) = max(k | pi(k) <= numOfZerofreeNum_n(k)), where numOfZerofreeNum_n(k) is the number of base-n zerofree numbers <= k ((see A324161 for general formulas regarding numOfZerofreeNum _n(k))).
%F a(n) >= A324154(n) + 1.
%F Estimate of the n-th term (n > 2):
%F a(n) < e*(p*log(p*log((e/(e-1))*p*log(p))))^(1/(1-d)),
%F a(n) > e^1.1*(q*log(q*log(q*log(q))))^(1/(1-d)),
%F where p := (n-1)/((n-2)*(1-d(n)))*e^(-(1-d)), q := (n-1)^d/((n-2)*(1-d(n)))*e^(-1.1*(1-d)), d := d(n) := log(n-1)/log(n).
%F Also, but more imprecise:
%F a(n) < e*((e/(e-1))*p*log(p))^(1/(1-d)),
%F a(n) < e*((e/(e-1))*p*log(p))^((n-1/2)*log(n)),
%F a(n) < n*((e/(e-1))*n*log(n)*log(n*log(n)))^((n-1/2)*log(n)), n > 3.
%F Asymptotic behavior:
%F a(n) = O(n*((e/(e-1))*n*log(n)*log(n*log(n)))^(n*log(n))).
%F a(n) = O(n*((e+1)/(e-1)*n*log(n)^2)^(n*log(n))).
%e a(2) = 4, since pi(1) = 0 <= 1 = numOfZerofreeNum_2(1), pi(2) = 1 <= 1 = numOfZerofreeNum_2(2), pi(3) = 2 <= 2 = numOfZerofreeNum_2(3), pi(4) = 2 <= 2 = numOfZerofreeNum_2(3), and pi(m) > numOfZerofreeNum_2(m) for m > 4, where numOfZerofreeNum_2(m) is the number of base-2-zerofree numbers <= m and pi(m) = number of primes <= m. The first base-2-zerofree numbers are 1 = 1_2, 3 = 11_2, 7 = 111_2, ...
%Y Cf. A011540, A052382, A324154, A324160, A324161, A325164, A325165.
%K nonn,base,more,hard
%O 2,1
%A _Hieronymus Fischer_, Feb 22 2019
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