A224504 a(n) = number of terms in row n of A214850.

2, 3, 6, 3, 3, 5, 3, 5, 4, 6, 4, 5, 4, 3, 8, 2, 3, 6, 3, 8, 6, 3, 8, 6, 2, 6, 8, 3, 4, 4, 4, 6, 4, 3, 8, 8, 2, 3, 7, 3, 8, 12, 3, 4, 8, 2, 8, 8, 3, 4, 8, 4, 8, 9, 4, 10, 6, 2, 5, 8, 2, 8, 6, 4, 6, 5, 3, 8, 4, 3, 8, 9, 4, 6, 7, 3, 8, 4, 4, 8, 5, 4, 8, 6, 5, 6

Offset: 1

Author

Michel Lagneau, Apr 08 2013

Keywords

• nonn

Comments

Number of multiplicative finite groups G(p) with elements {T(2n+1,k)/pZ} where T(2n+1,k) is the reduced trajectory of the Collatz problem whose elements are all odd and p <= A075684(n) + 1.

Examples

			a(18) = 6 because there exist 6 finite groups given by row 18 of A214850 where p = 2, 4, 6, 8, 12 and 18. The Collatz trajectory of the number 2*18 + 1 = 37 with odd numbers is T(37,k) = {37, 7, 11, 17, 13, 5, 1}, and the 6 groups G(p) are:
G(2) = {T(37,k)/2Z} = {1}
G(4) = {T(37,k)/4Z} = {1, 3}
G(6) = {T(37,k)/6Z} = {1, 5}
G(8) = {T(37,k)/8Z}  = {1, 3, 5, 7}
G(12) = {T(37,k)/12Z} = {1, 5, 7, 11}
G(18) = {T(37,k)/18Z} = {1, 5, 7, 11, 13, 17}
G(18) is a cyclic group because the element 5 (or 11) generates the group:
5^1 == 5, 5^2 == 7, 5^3 == 17, 5^4 == 13, 5^5 == 11, 5^6 == 1 (mod 18).
G(8) is not a cyclic group.
a(170) = 32 because there exist 32 finite groups with two elements given by row 170 of A214850 where p = 2, 4, 6, 8, 10, 12, 18, 20, 24, 30, 34, 36, 38, 40, 60, 68, 72, 76, 90, 102, 114, 120, 136, 152, 170, 180, 190, 204, 228, 306, 340, 342. The Collatz trajectory of the number 2*170 + 1 = 341 with odd numbers is T(341,k) = {1, 341}.
		

Links

• Michel Lagneau, Table of n, a(n) for n = 1..500

Crossrefs

Cf. A075680, A075684, A214850.