A324165 -id:A324165 - cp's OEIS frontend

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

2, 3, 344251, 33182655683

Offset: 2

Hieronymus Fischer, Feb 22 2019

Further terms:
4.1645275173242*10^15  < a(6)  < 4.164601237609*10^15,
1.0163657136*10^22     < a(7)  < 1.0163715977928*10^22,
8.4513797224747*10^28  < a(8)  < 8.4514006058085*10^28,
1.959502408617*10^36   < a(9)  < 1.9595048275153*10^36,
1.0953002073198*10^44  < a(10) < 1.0953009588121*10^44,
1.3480809599483*10^53  < a(11) < 1.3480814844466*10^53,
3.540916347013*10^61   < a(12) < 3.5409172310273*10^61,
2.080341784427*10^71   < a(13) < 2.0803421176765*10^71,
2.4843833393543*10^81  < a(14) < 2.4843836067277*10^81,
5.6615671922884*10^91  < a(15) < 5.6615676172791*10^91,
2.1556069128839*10^148 < a(20) < 2.1556069510899*10^148.
a(n) is always a prime number.
For n > 2, all terms are odd.
All terms a(n) are zero-containing numbers (in base n), a(n) - 1 is also a zero-containing number (in base n).
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.
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.
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.

			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, ...
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, ...

Cf. A011540, A052382, A324155, A324160, A324161, A324164, A324165.

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

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))).
a(n) <= A324155(n) - 1.
Estimation for the n-th term (n > 2):
a(n) < e*(p*log(p*log((e/(e-1))*p*log(p))))^(1/(1-d)),
a(n) > e^1.1*(q*log(q*log(q*log(q))))^(1/(1-d)),
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).
Also, but more imprecise:
a(n) > e^1.1*(q*log(q))^(1/(1-d)),
a(n) > (n/(n-1))*((n-1)*log(n)*log(n*log(n)))^((n-1/2))*log(n)).
Asymptotic behavior:
a(n) = O(n*((e/(e-1))*n*log(n)*log(n*log(n)))^(n*log(n))).
a(n) = O(n*(((e+1)/(e-1))*n*log(n)^2)^(n*log(n))).

A324164 Number of primes <= A324154(n).

Original entry on oeis.org

1, 2, 29523, 1431655764, 119209289550780, 204698073815493849906, 1288498953284574087356182400, 23736214210444926301853697505006152, 1090995446010964053236424684934590917505180, 1111111111111111111111111111111111111111111111111110

Offset: 2

Hieronymus Fischer, Feb 22 2019

nonn

Also the number of zerofree numbers <= A324154(n).
Expressed in base n - 1 and starting with n = 3, the sequence is 10, 1111111110, 1111111111111110, 111111111111111111110, 111111111111111111111111110, 111111111111111111111111111111110, 111111111111111111111111111111111111110, 111111111111111111111111111111111111111111110, 1111111111111111111111111111111111111111111111111110, ....
Ostensibly, the reason for that is the calculation formula (see Formula section) for the number of zerofree numbers <= x^m + y, with y < (x^(m+1)-1)/(x-1) - x^m. But the deeper reason is the definition of sequence A324154. Each term A324154(n) marks a point of intersection between the curve numOfZerofreeNum_n(x) [the number of base-n zerofree numbers <= x] and the curve pi(x) [the number of prime numbers <= x]. Since numOfZerofreeNum_n(x) doesn't change for relatively large intervals at x = k*n^m (approx. a portion of > 1/(k*n)), but grows similar to pi(x) for regions outside, it is likely, that the point of intersection lies between x = k*n^m and x = n^m*(k + 1/n + 1/n^2 + 1/n^3 + ... + 1/n^m). The chance is maximal for k = 1, since the density of primes becomes smaller for greater x.

			a(2) = 1, since there is only one prime <= A324154(2) = 2.
a(3) = 2, since there are 2 primes <= A324154(3) = 3.

Cf. A011540, A052382, A324154, A324155, A324160, A324161, A324165.

a(n) = pi(A324154(n)).
a(n) = numOfZerofreeNum_n(A324154(n)), where numOfZerofreeNum_n(x) is the number of base-n zerofree numbers <= x (cf. A324161).
a(n) = k*(n-1)^m + ((n-1)^m - 1)/(n-2) - 1,
where m = floor(log_n(A324154(n))), k = floor(A324154(n)/n^m), and provided A324154(n) - k*n^m < (n^(m+1)-1)/(n-1) - n^m.
With d := log(n-1)/log(n):
a(n) <= ((n - 1)*(A324154(n) + 1)^d - 1)/(n - 2) - 1,
a(n) >= (((n - 1)*A324154(n) + n)^d - 1)/(n - 2) - 1.
a(n) < A324154(n) / (log(A324154(n)) - 1.1), for n > 3.
a(n) > A324154(n) / (log(A324154(n)) - 1), for n > 3.

cp's OEIS Frontend

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

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