|
|
A361899
|
|
a(n) = 3*(6858365065530*(2^45 - 1)*n + 153479820268467961)^2.
|
|
3
|
|
|
70668165688923686196507258250492563, 174687593550891106640307045856561008882907291372256643, 698750373759134872171732581703201135992894186495330123, 1572188340624731296664944773228844067526467943619713003
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,1
|
|
COMMENTS
|
The "1/k" heuristic predicts that primes of the form k*2^m + 1 with k odd and m > 0 have almost a 1/k chance of being Fermat divisors (Dubner and Keller). This sequence yields a correction to the "1/k" heuristic, because it generates special values of k.
If:
1) k is of the form 3*a^2, where a is an odd positive integer not divisible by 3,
2) k is not a Sierpiński number,
3) for all odd positive integers m the numbers k*2^m + 1 are composite,
then the probability that a Fermat number is divisible by a prime of the form k*2^m + 1 equal to 0.
Every term meets the first and third condition. For any n, at least one of the primes from A361898 (except 3) divides every integer in the sequence a(n)*2^m + 1 with m odd.
What is the smallest odd integer k such that every prime of the form k*2^m + 1 (m > 0) does not divide any Fermat number?
|
|
REFERENCES
|
H. Suyama, A note on the factors of Fermat numbers II, Abstracts of Papers Presented to the Amer. Math. Soc., Vol. 5 (1984), p. 132.
|
|
LINKS
|
|
|
FORMULA
|
G.f.: (70668165688923686196507258250492563 + 174687593550891106428302548789789950293385516620778954*x + 174687593106461552462815941200289167933694087130037883*x^2)/(1 - x)^3.
a(n) = 3*(2*(Product_{i=1..13} A361898(i))*n + 153479820268467961)^2.
a(n) = 3*((29062/1192737)*(2^48 - 1)*(2^45 - 1)*n + 153479820268467961)^2.
|
|
MATHEMATICA
|
Table[3 (6858365065530 (2^45 - 1) n + 153479820268467961)^2, {n, 0, 3}]
|
|
PROG
|
(Magma) [3*(6858365065530*(2^45-1)*n+153479820268467961)^2: n in [0..3]];
(PARI) a(n)=3*(6858365065530*(2^45-1)*n+153479820268467961)^2
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|