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

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, 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, 7, 4, 15, 2, 8, 5, 13, 2, 17, 2, 7, 9, 8, 4, 16, 2, 13, 5, 8, 2

Offset: 1

Author

Michael De Vlieger, Jul 16 2024

Keywords

• nonn

Comments

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.

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.

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.

Observation: A008479(n) < a(n) <= A010846(n).

Examples

			Let b = A010846.
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.
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.
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.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Comparison of S(900) in red with R(900) in blue
• Eric Weisstein's World of Mathematics, Simplex.

Crossrefs

Cf. A001221, A008479, A010846, A162306, A246655.

Programs

Mathematica

{1}~Join~Table[Floor[(1/PrimeNu[n]!)*Times @@ Map[Log[#, n] + 1 &, FactorInteger[n][[All, 1]] ] ], {n, 2, 82}]

Formula

a(n) = A010846(n) = A008479(n) + 1 = 2 for n such that omega(n) = 1.