|
|
A296240
|
|
Pisano quotients: a(n) = (p-1)/k(p) if p == +- 1 mod 5, = (2*p+2)/k(p) if p == +- 2 mod 5, where p = prime(n) and k(p) = Pisano period(p).
|
|
8
|
|
|
1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 9, 5, 1, 1, 2, 9, 1, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 7, 1, 1, 1, 3, 1, 3, 2, 3, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 5, 1, 1, 1, 1, 1, 1, 10, 1, 1, 1, 1, 1, 1, 1, 2, 20, 1, 6, 1, 9, 3, 1, 1, 1, 1, 1, 1
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
4,7
|
|
COMMENTS
|
Wall (1960) in Theorems 6 and 7 proved that a(n) is an integer for n >= 4. Jarden (1946) proved that the sequence is unbounded. See Elsenhans and Jahnel (2010), pp. 1-2.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = (3 - L(p))/2 * (p - L(p)) / k(p), where p = prime(n), L(p) = Legendre(p|5), and k(p) = Pisano period(p) = A001175(p).
a(n) > 1 if and only if prime(n) is in A222413.
|
|
MATHEMATICA
|
With[{p = Prime[n]}, T = Table[a = {1, 0}; a0 = a; k = 0; While[k++; s = Mod[Plus @@ a, p]; a = RotateLeft[a]; a[[2]] = s; a != a0]; k, {n, 1, 130}]; Table[L = KroneckerSymbol[p, 5]; (3 - L)/2 (p - L)/T[[n]], {n, 4, 130}]] (* after T. D. Noe *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|