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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.

%I A324156 #7 Aug 16 2019 00:15:03
%S A324156 2,3,4,3,113,114,115,116,117,118,119,120,199,200,201,293,294,295,296,
%T A324156 297,298,299,300,301,302,303,304,305,306,482,483,491,492,493,494,495,
%U A324156 496,497,498,344251,344252,344253,344254,344255,344256,351902,353501,353502,353503
%N 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.
%C A324156 The offset is 2 since the least base (= row) for which the definition makes sense is n = 2.
%C A324156 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.
%C A324156 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.
%e A324156 T(2,1) = 2, since pi(2) = 1 = numOfZerofreeNum_2(2) where numOfZerofreeNum_n(k) = number of base-n zerofree numbers <= k.
%e A324156 T(2,2) = 3, since pi(3) = 2 = numOfZerofreeNum_2(3).
%e A324156 T(3,2) = 4, since pi(4) = 2 = numOfZerofreeNum_2(3).
%e A324156 T(3,1) = 3, since pi(3) = 2 = numOfZerofreeNum_3(3).
%e A324156 T(3,2) = 113, since pi(113) = 30 = numOfZerofreeNum_3(113).
%e A324156 Triangle T(n,k) begins:
%e A324156 2, 3, 4;
%e A324156 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;
%e A324156 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;
%e A324156 33182655683, 33182655684, 33182655685, 33182655686, 33182655687, 33182655688;
%Y A324156 Cf. A324160, A324161, A324154, A324155, A324157.
%K A324156 nonn,base,tabf
%O A324156 2,1
%A A324156 _Hieronymus Fischer_, Jul 16 2019