cp's OEIS Frontend

A324154 Least number N such that the number of primes (<= N) >= the number of the base-n-zerofree numbers (<= N).

%I A324154 #28 Mar 28 2019 12:00:38
%S A324154 2,3,344251,33182655683
%N A324154 Least number N such that the number of primes (<= N) >= the number of the base-n-zerofree numbers (<= N).
%C A324154 Further terms:
%C A324154 4.1645275173242*10^15  < a(6)  < 4.164601237609*10^15,
%C A324154 1.0163657136*10^22     < a(7)  < 1.0163715977928*10^22,
%C A324154 8.4513797224747*10^28  < a(8)  < 8.4514006058085*10^28,
%C A324154 1.959502408617*10^36   < a(9)  < 1.9595048275153*10^36,
%C A324154 1.0953002073198*10^44  < a(10) < 1.0953009588121*10^44,
%C A324154 1.3480809599483*10^53  < a(11) < 1.3480814844466*10^53,
%C A324154 3.540916347013*10^61   < a(12) < 3.5409172310273*10^61,
%C A324154 2.080341784427*10^71   < a(13) < 2.0803421176765*10^71,
%C A324154 2.4843833393543*10^81  < a(14) < 2.4843836067277*10^81,
%C A324154 5.6615671922884*10^91  < a(15) < 5.6615676172791*10^91,
%C A324154 2.1556069128839*10^148 < a(20) < 2.1556069510899*10^148.
%C A324154 a(n) is always a prime number.
%C A324154 For n > 2, all terms are odd.
%C A324154 All terms a(n) are zero-containing numbers (in base n), a(n) - 1 is also a zero-containing number (in base n).
%C A324154 The numbers between p := max( k < a(n) | k is prime) and a(n) + 1 are zero-containing numbers, i.e., a(n) + 1 - j is a zero-containing number for 0 < j < a(n) + 1 - p.
%C A324154 From the equality A324164(5) = A324165(5) we can conclude that a(5) and A324155(5) + 1 are proximate primes. Same is true for a(11): a(11) and A324155(11) + 1 are proximate primes.
%C A324154 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 A324154 a(n) = min(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 A324154 a(n) <= A324155(n) - 1.
%F A324154 Estimation for the n-th term (n > 2):
%F A324154 a(n) < e*(p*log(p*log((e/(e-1))*p*log(p))))^(1/(1-d)),
%F A324154 a(n) > e^1.1*(q*log(q*log(q*log(q))))^(1/(1-d)),
%F A324154 where p := (n-1)/((n-2)*(1-d))*e^(-(1-d)), q := (n-1)^d/((n-2)*(1-d))*e^(-1.1*(1-d)), d := d(n) := log(n-1)/log(n).
%F A324154 Also, but more imprecise:
%F A324154 a(n) > e^1.1*(q*log(q))^(1/(1-d)),
%F A324154 a(n) > (n/(n-1))*((n-1)*log(n)*log(n*log(n)))^((n-1/2))*log(n)).
%F A324154 Asymptotic behavior:
%F A324154 a(n) = O(n*((e/(e-1))*n*log(n)*log(n*log(n)))^(n*log(n))).
%F A324154 a(n) = O(n*(((e+1)/(e-1))*n*log(n)^2)^(n*log(n))).
%e A324154 a(2) = 2, since pi(1) = 0 < 1 = numOfZerofreeNum_2(1), pi(2) = 1 >= 1 = numOfZerofreeNum_2(2), 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, ...
%e A324154 a(3) = 3, since pi(1) = 0 < 1 = numOfZerofreeNum_3(1), pi(2) = 1 < 2 = numOfZerofreeNum_3(2), pi(3) = 2 >= 2 = numOfZerofreeNum_3(3), where numOfZerofreeNum_3(m) is the number of base-3-zerofree numbers <= m and pi(m) = number of primes <= m. The first base-3-zerofree numbers are 1 = 1_3, 2 = 2_3, 4 = 11_3, 5 = 12_3, 7 = 21_3, ...
%o A324154 (PARI) a(n) = {my(k = 1, nbp = 0, nzf = 1); while(nbp < nzf, k++; if (isprime(k), nbp++); if (vecmin(digits(k, n)), nzf++);); k;} \\ _Michel Marcus_, Mar 20 2019
%Y A324154 Cf. A011540, A052382, A324155, A324160, A324161, A324164, A324165.
%K A324154 nonn,base,hard,more
%O A324154 2,1
%A A324154 _Hieronymus Fischer_, Feb 22 2019