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A384000 Smallest number k with n distinct prime factors such that A010846(k) = A024718(n) (a tight lower bound), or -1 if such k does not exist.

%I A384000 #13 Jun 11 2025 01:07:55
%S A384000 1,2,6,1001,268801,3433936673,2603508937756211
%N A384000 Smallest number k with n distinct prime factors such that A010846(k) = A024718(n) (a tight lower bound), or -1 if such k does not exist.
%C A384000 These numbers k have the smallest A010846(k) for a number with n distinct prime factors.
%C A384000 a(7) <= 1398483454696343742813089 = 1049 * 2819 * 3319 * 3433 * 3457 * 3463 * 3467.
%C A384000 a(8) <= 32829974457045619959776094471833047127947.
%e A384000 Table of a(n), n = 0..6, showing prime decomposition and cardinality of row a(n) of A162306, c(n) = A010846(a(n)) = A024718(n).
%e A384000 n               a(n)   c(n)    prime factors of a(n)        a(n)
%e A384000 ----------------------------------------------------------------------
%e A384000 0                  1     1     -
%e A384000 1                  2     2     2                            A000040(1)
%e A384000 2                  6     5     2,   3                       A138109(1)
%e A384000 3               1001    15     7,  11,  13                  A383177(1)
%e A384000 4             268801    50    13,  23,  29,  31             A383178(2)
%e A384000 5         3433936673   176    41,  83,  97, 101, 103        A383179(209)
%e A384000 6   2603508937756211   638   163, 373, 439, 457, 461, 463
%e A384000 Tables of terms m in r(a(n)) = row a(n) of A162306, writing instead only exponents i of prime power factors p^i | m for  each p | a(n), written in order of the prime base:
%e A384000 For n = 2, i.e., squarefree semiprime k in A138109 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-2 table:
%e A384000   00  10  20
%e A384000   01  11
%e A384000 Thus row 6 of A162306 has the following elements:
%e A384000    1   2   4
%e A384000    3   6
%e A384000 For n = 3, i.e., sphenic k in A383177 (that achieves the lower bound), we have the following ordered exponent combinations in a rank-3 table:
%e A384000   000 100 200 300     001 101 201     002
%e A384000   010 110 210         011 111
%e A384000   020 120
%e A384000 Thus row 1001 of A162306 has the following elements:
%e A384000     1   7  49 343      13   91 637    169
%e A384000    11  77 539         141 1001
%e A384000   121 857
%Y A384000 Cf. A001700, A001221, A005117, A007947, A010846, A024718, A138109, A162306, A383177, A383178, A383179.
%K A384000 nonn,hard,more
%O A384000 0,2
%A A384000 _Michael De Vlieger_, May 19 2025