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A373738 a(1) = 1, a(n) = floor((1/omega(n)!) * Product_{p | n} 1 + (log n)/(log p)), where omega = A001221.

%I A373738 #13 Aug 29 2024 17:47:06
%S A373738 1,2,2,3,2,4,2,4,3,5,2,7,2,5,4,5,2,9,2,7,4,6,2,10,3,6,4,7,2,12,2,6,5,
%T A373738 6,4,13,2,6,5,10,2,13,2,8,7,7,2,14,3,11,5,8,2,15,4,10,5,7,2,19,2,7,7,
%U A373738 7,4,15,2,8,5,13,2,17,2,7,9,8,4,16,2,13,5,8,2
%N A373738 a(1) = 1, a(n) = floor((1/omega(n)!) * Product_{p | n} 1 + (log n)/(log p)), where omega = A001221.
%C A373738 This sequence is the integer part of the omega(n)-rank content of an omega(n)-rank orthogonal simplex S(n) with axes measuring 1 + (log n)/(log p) for all primes p | n.
%C A373738 Let R(n) be the arrangement of row n of A162306(n) according to the order of exponents of distinct prime factors p | n. Then A010846(n) is the content of a rank omega(n) Hauy construction where the numbers are placed in omega(n) dimensional cubes, while S(n) is the corresponding simplex.
%C A373738 Conjecture: A010846(k) - a(k) approaches 0 as k increases toward infinity, for k with omega(k) > 1 that have the same squarefree kernel r. Therefore, the difference is most significant for composite squarefree k.
%C A373738 Observation: A008479(n) < a(n) <= A010846(n).
%H A373738 Michael De Vlieger, <a href="/A373738/b373738.txt">Table of n, a(n) for n = 1..10000</a>
%H A373738 Michael De Vlieger, <a href="/A373738/a373738.png">Comparison of S(900) in red with R(900) in blue</a>
%H A373738 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Simplex.html">Simplex</a>.
%F A373738 a(n) = A010846(n) = A008479(n) + 1 = 2 for n such that omega(n) = 1.
%e A373738 Let b = A010846.
%e A373738 a(6) = 4 since the floor of the area of a right triangle with axial edge lengths {1+log_p 6 : p | 6} = {3.58496..., 2.63093...}, a(6) = floor(9.43178.../2) = 4. b(6) = 5.
%e A373738 a(10) = 5 since the floor of the area of a right triangle with axial edge lengths {1+log_p 12 : p | 12} = {4.32193..., 2.43068...}, a(10) = floor(10.5052.../2) = 5. b(10) = 6.
%e A373738 a(30) = 12 since the floor of the volume of a trirectangular tetrahedron with axial edge lengths {1+log_p 30 : p | 30} = {5.90689..., 4.0959..., 3.11328...}, a(30) = floor(75.3229.../6) = 12. b(30) = 18.
%e A373738 a(210) = 34 since the floor of the content of a 4-simplex with a vertex with orthogonal edges at origin and axial edge lengths {1+log_p 210 : p | 210} = {8.71425..., 5.86715..., 4.32234..., 3.74787...}, a(210) = floor(828.248.../24) = 12. b(210) = 68, etc.
%t A373738 {1}~Join~Table[Floor[(1/PrimeNu[n]!)*Times @@ Map[Log[#, n] + 1 &, FactorInteger[n][[All, 1]] ] ], {n, 2, 82}]
%Y A373738 Cf. A001221, A008479, A010846, A162306, A246655.
%K A373738 nonn
%O A373738 1,2
%A A373738 _Michael De Vlieger_, Jul 16 2024