OFFSET
1,4
LINKS
Max Alekseyev, Table of n, a(n) for n = 1..10000
R. J. Mathar, List of element order statistics for n <= 64.
FORMULA
For all squarefree n, a(n)=0, since there is only one abelian group of order n. Hence the group is trivially known without any checking.
EXAMPLE
For n=20, by the fundamental theorem of finite abelian groups, the group is either Z20 or Z10 x Z2. At worst, you could choose the identity, 1 element of order 2, 4 elements of order 5, and 4 elements of order 10. Then you still wouldn't know which group you have. But the order of the next element you choose will determine the group you have. So a(20)=11.
The previous value was a(16) = 9; It should be 13. Two of the size-16 groups have shapes [4,2,2] and [4,4], with element-orders:quantities
[4,2,2] 1:1 2:7 4:8
[4,4] 1:1 2:3 4:12
The sample 1:1, 2:3, 4:8 (12 in total) won't distinguish those two. - Don Reble, Oct 04 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Isaac Lambert, Oct 10 2010
EXTENSIONS
Corrected and extended by Don Reble - N. J. A. Sloane, Oct 04 2023
a(1)=0 prepended by Max Alekseyev, Oct 07 2023
STATUS
approved