cp's OEIS Frontend

A306521 Least integer N > 2 such that the number of primes (<=N) <= the number of base-n-zero containing numbers (<=N).

%I A306521 #38 Aug 25 2019 13:32:12
%S A306521 3,3,4,28,42,104,136,329,510,856,1449,2212,2782,3434,4188,5042,6001,
%T A306521 7082,8276,9604,11062,12666,14405,31651,35694,40061,66427,73966,
%U A306521 108764,149756,197516,288280,354924,515538,701002,963687,1318399,1840377
%N A306521 Least integer N > 2 such that the number of primes (<=N) <= the number of base-n-zero containing numbers (<=N).
%C A306521 a(n) <= A306526(n), equality holds for n = 2, 14, 15, 16, 17, 18, 19, 20, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52 (but  a(n) < A306526(n) for all other indices n up to 82). For sufficiently large n, equality holds true for those bases n which satisfy 1/2 <= fract(sqrt(n/log(n)) + O(sqrt(log(n)/n))) < 3/4. This is true for infinitely many further indices, at least for all bases n = ceiling(x), where x is a solution of x/log(x) = k-th triangular number + 1/4, k > 1. For k = 2..10 the corresponding bases are n = 19, 48, 92, 152, 230, 326, 440, 574, 727. Let e(n) be the number of bases m <= n for which a(m) = A306526(m), then lim_{n->infinity} e(n)/n >= 1/4. Conjecture: lim_{n->infinity} e(n)/n = 1/4.
%H A306521 Hieronymus Fischer, <a href="/A306521/b306521.txt">Table of n, a(n) for n = 2..100</a>
%F A306521 With numOfZeroNum_n(k) [= the number of base-n-zero containing numbers <= k] and pi(k) [= the number of primes <= k] and d := log(n-1)/log(n):
%F A306521 a(n) = min(k > 2 | pi(k) <= numOfZeroNum_n(k)). Because of d = d(n) < 1, numOfZeroNum_n(k) = k*(1 + O(k^(d-1)), pi(k) = k/log(k)*(1+o(1)), and pi(3) = 2 >= 2 = numOfZeroNum_n(3), this minimum always exists (for n > 2). The case n = 2 is obvious. See A324160 regarding general formulas for numOfZeroNum_n(k).
%F A306521 Estimate of the n-th term:
%F A306521 a(n) > e*(1 + c1/c2*(1 + sqrt(1 + c2*c3/c1^2)))^(1/(1-d)), for n > 6,
%F A306521 where d := log(n-1)/log(n),
%F A306521 c0 := e^(1-d),
%F A306521 c1 := (n-1)^d/(n-2) - 1/e^(sqrt(n*log(n))) - d*c0,
%F A306521 c2 := (n-1)^d/(n-2) - 1/e^(sqrt(n*log(n))) + (1 - 1/sqrt(n*log(n)))*c0,
%F A306521 c3 := 2*(1-d)*c0.
%F A306521 Also, but less accurate, for n > 6,
%F A306521 a(n) > e*(1 + 1/(sqrt(n*log(n)) - 2))^(1/(1-d)).
%F A306521 a(n) > e*(1 + 1/(sqrt(n*log(n)) - 2))^((n-1/2)*log(n)).
%F A306521 a(n) <= A306526(n), see A306526 for further upper bound estimations.
%F A306521 Asymptotic behavior:
%F A306521 a(n) = O(sqrt(n)*e^sqrt(n*log(n))).
%F A306521 lim sup a(n)/e^(sqrt(n*log(n))+(log(n)+1)/2) = 1, for n --> infinity.
%F A306521 lim inf a(n)/e^(sqrt(n*log(n))+1/2) = 1, for n --> infinity.
%e A306521 a(2) = 3, since pi(3) = 2 <= 2 = numOfZeroNum_2(3), where numOfZeroNum_2(m) is the number of base-2-zero containing numbers <= m and pi(m) = number of primes <= m. The first base-2-zero containing numbers are 0 = 0_2, 2 = 10_2, 4 = 100_2, ... (Hint: numbers <= 2 are out of scope for self-evident reasons).
%e A306521 a(3) = 3, since pi(3) = 2 <= 2 = numOfZeroNum_3(3), where numOfZeroNum_3(m) is the number of base-3-zero containing numbers <= m and pi(m) = number of primes <= m. The first base-3-zero containing numbers are 0 = 0_2, 3 = 10_3, 6 = 20_3, 9 = 100_3, 10 = 101_3, 11 = 102_3, 12 = 120_3, ...
%e A306521 a(4) = 4, since pi(3) = 2 > 1 = numOfZeroNum_4(3), pi(4) = 2 <= 2 = numOfZeroNum_4(4), where numOfZeroNum_4(m) is the number of base-4-zero containing numbers <= m and pi(m) = number of primes <= m. The first base-4-zero containing numbers are 0 = 0_2, 4 = 10_4, 8 = 20_4, ...
%o A306521 (PARI) cz(m,n) = vecmin(digits(m, n))==0;
%o A306521 a(n) = {my(m=2, nbz=1+sum(k=1, 2, cz(k,n)), pmp=primepi(2)); for (m=3, oo, if (isprime(m), pmp++); if (cz(m,n), nbz++); if (pmp <= nbz, return (m)););} \\ _Michel Marcus_, Jun 10 2019
%Y A306521 Cf. A011540, A052382, A306195, A306442, A306526, A324154, A324155, A324160, A324161.
%K A306521 nonn,base
%O A306521 2,1
%A A306521 _Hieronymus Fischer_, Mar 29 2019