Search: a212203 -id:a212203
|
| |
| |
|
|
|
1, 2, 4, 8, 16, 18, 65, 72, 256, 258, 260, 266, 4101, 4132, 4167, 4290, 65536, 65540, 65542, 65544, 65594, 65600, 65658, 65694
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
REFERENCES
|
J. H. Conway and Alex Ryba, Personal communication, May 03 2012 and Jun 10 2012
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
A212200 begins 1, 3, 3, 15, 15, 15, 15, 5, 15, 5, 15, 15, 5, 5, 15, 85, ..., containing 1, 3, 15, 5, 85, ..., which gives the beginning of A212203. The numbers in question appear at positions 1, 2, 4, 8, 16, 18, ..., which is A212204.
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
More terms from Alex Ryba, Jun 10 2012
|
|
|
STATUS
|
approved
|
| |
|
|
| |
|
|
A212200
|
|
Multiplicative order of n in nim-multiplication.
|
|
+10
9
|
|
|
|
1, 3, 3, 15, 15, 15, 15, 5, 15, 5, 15, 15, 5, 5, 15, 85, 85, 255, 255, 85, 85, 255, 255, 85, 85, 255, 255, 255, 255, 85, 85, 255, 255, 255, 255, 85, 255, 85, 255, 255, 255, 255, 255, 255, 85, 255, 85, 255, 85, 85, 255, 255, 255, 255, 255, 255, 255, 255, 255, 255, 85, 85, 255, 255, 51, 255, 255, 255, 51, 255, 255, 17, 255, 85, 255, 17, 255, 85
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
The observation above is incorrect. Note that {0,1,...,2^2^k-1} together with the nim operations makes a field isomorphic to GF(2^2^k). This means that:
- Every number is a divisor of a number of the form 2^2^k-1, and every divisor of 2^2^k-1 for some k appears;
- If d is a divisor of 2^2^k-1 for some k, then d appears phi(d) times among {a(1),a(2),...,a(2^2^m-1)} for all m >= k, phi = A000010. This means that if d > 1, and k is the smallest number such that d | 2^2^k-1, then d can only appear among {a(2^2^(k-1)),...a(2^2^k-1)}.
So the correct result should be: all terms are divisors of numbers of the form 2^2^k-1, and each divisor d appears phi(d) times.
For example, 641 would appear 640 times in this sequence, among {a(2^32),...,a(2^64-1)}, although to determine their positions is hard. (End)
|
|
|
REFERENCES
|
J. H. Conway, On Numbers and Games, Academic Press, Chapter 6.
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
The nim-products 4*4*...*4 are (cf. A051775): 4, 4^2=6, 4^3=4*6=14, 4^4=4*14=5, 4^5=2, 4^6=8, ..., 4^14=15, 4^15=1, so 4 has order a(4) = 15.
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
STATUS
|
approved
|
| |
|
|
| |
| |
| |
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
a(7)-a(9) from Alex Ryba, Jun 10 2012
|
|
|
STATUS
|
approved
|
| |
|
|
| |
| |
| |
|
|
|
OFFSET
|
1,2
|
|
|
LINKS
|
|
|
|
CROSSREFS
|
|
|
|
KEYWORD
|
nonn,more
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
a(7)-a(9) from Alex Ryba, Jun 10 2012
|
|
|
STATUS
|
approved
|
| |
|
Search completed in 0.010 seconds
|
