A324160 -id:A324160 - cp's OEIS frontend

A324165 The number of primes <= A324155(n).

2, 94, 88572, 1431655764, 405311584472655, 375279801995072058162, 2392926627528494733661481601, 44505401644584236815975682821886536, 9818959014098676479127822164411318257546629, 1111111111111111111111111111111111111111111111111110

Offset: 2

Hieronymus Fischer, Mar 05 2019

nonn

Also the number of zerofree numbers <= A324155(n).
Expressed in base n - 1 and starting with n = 3, the sequence is 1011110, 11111111110, 1111111111111110, 411111111111111111110, 211111111111111111111111110, 211111111111111111111111111111110, 211111111111111111111111111111111111110, 1111111111111111111111111111111111111111111110, 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 A324155. Each term A324155(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. Nevertheless, k > could also happen as we can see for n = 6, 7, 8 and 9.

			a(2) = 2, since there are 2 primes <= A324155(2) = 4.
a(3) = 94, since there are 94 primes <= A324155(3) = 498.

Cf. A011540, A052382, A324154, A324155, A324160, A324161, A324164.

a(n) = pi(A324155(n)).
a(n) = numOfZerofreeNum_n(A324155(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(A324155(n))), k = floor(A324155(n)/n^m), and provided A324155(n) - k*n^m < (n^(m+1)-1)/(n-1) - n^m.
With d := log(n-1)/log(n):
a(n) <= ((n - 1)*(A324155(n) + 1)^d - 1)/(n - 2) - 1.
a(n) >= (((n - 1)*A324155(n) + n)^d - 1)/(n - 2) - 1.
a(n) < A324155(n) / (log(A324155(n)) - 1.1), for n > 3.
a(n) > A324155(n) / (log(A324155(n)) - 1), for n > 3.

A324156 Irregular triangle T(n,k) read by rows in which row n lists the numbers m such that the number of prime numbers <= m is equal to the number of base-n zerofree numbers <= m.

Original entry on oeis.org

2, 3, 4, 3, 113, 114, 115, 116, 117, 118, 119, 120, 199, 200, 201, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 482, 483, 491, 492, 493, 494, 495, 496, 497, 498, 344251, 344252, 344253, 344254, 344255, 344256, 351902, 353501, 353502, 353503

Offset: 2

Hieronymus Fischer, Jul 16 2019

The offset is 2 since the least base (= row) for which the definition makes sense is n = 2.
The least term of row n is T(n,1) = A324154(n). The last term of row n is T(n,j) = A324155(n), where j = A324157(n) is the number of terms of the n-th row.
Terms of rows higher than 5 are unknown, but they are bounded by the above rule. For example, the first term of the 6th row is T(6,1) = A324154(n) = 4.1645*10^15, approximately. The last term of the 6th row is A324155(n) = 1.46705*10^16, approximately.

			T(2,1) = 2, since pi(2) = 1 = numOfZerofreeNum_2(2) where numOfZerofreeNum_n(k) = number of base-n zerofree numbers <= k.
T(2,2) = 3, since pi(3) = 2 = numOfZerofreeNum_2(3).
T(3,2) = 4, since pi(4) = 2 = numOfZerofreeNum_2(3).
T(3,1) = 3, since pi(3) = 2 = numOfZerofreeNum_3(3).
T(3,2) = 113, since pi(113) = 30 = numOfZerofreeNum_3(113).
Triangle T(n,k) begins:
2, 3, 4;
3, 113, 114, 115, 116, 117, 118, 119, 120, 199, 200, 201, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 482, 483, 491, 492, 493, 494, 495, 496, 497, 498;
344251, 344252, 344253, 344254, 344255, 344256, 351902, 353501, 353502, 353503, 353504, 353505, 353506, 353507, 353508, 353509, 353510, 353511, 353512, 353513, 353514, 353515, 353516, 353517, 353518, 353519, 353520, 353521, 353522, 353523, 353524, 353525, 353526, 353631, 353632, 353633, 353634, 353635, 353636, 601379, 601380, 601381, 601382, 601383, 601384, 601385, 601386, 601387, 601388, 601389, 601390, 601391, 601392, 601393, 601394, 601395, 601396, 617903, 617904, 617905, 617906, 617907, 617908, 867281, 867282, 867283, 867284, 867285, 867286, 867287, 867288, 867289, 867290, 867291, 867292, 867293, 867294, 867295, 867296, 867297, 867298, 867299, 867300, 876414, 876431, 876432, 876437, 877213, 877214, 877215, 877216, 877217, 877218, 877219, 877220, 877221, 877222, 878014, 878021, 878022, 878037, 1139549, 1139550, 1139551, 1139552, 1139553, 1139554, 1139555, 1139556;
33182655683, 33182655684, 33182655685, 33182655686, 33182655687, 33182655688;

Cf. A324160, A324161, A324154, A324155, A324157.

cp's OEIS Frontend

A324165 The number of primes <= A324155(n).

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