|
|
A039956
|
|
Even squarefree numbers.
|
|
50
|
|
|
2, 6, 10, 14, 22, 26, 30, 34, 38, 42, 46, 58, 62, 66, 70, 74, 78, 82, 86, 94, 102, 106, 110, 114, 118, 122, 130, 134, 138, 142, 146, 154, 158, 166, 170, 174, 178, 182, 186, 190, 194, 202, 206, 210, 214, 218, 222, 226, 230, 238, 246, 254, 258, 262
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Sum of even divisors = 2 * the sum of odd divisors. - Amarnath Murthy, Sep 07 2002
a(n) = n * (3/1) * zeta(2) + O(n^(1/2)) = n * (3/1) * (Pi^2 / 6) + O(n^(1/2)).
For any prime p_i, the n-th squarefree number even to p_i (divisible by p_i) is:
n * ((p_i + 1)/1) * zeta(2) + O(n^(1/2)) = n * ((p_i + 1)/1) * (Pi^2 / 6) + O(n^(1/2)).
For any prime p_i, there are as many squarefree numbers having p_i as a factor as squarefree numbers not having p_i as a factor amongst all the squarefree numbers (one-to-one correspondence, both cardinality aleph_0).
E.g., there are as many even squarefree numbers as there are odd squarefree numbers.
For any prime p_i, the density of squarefree numbers having p_i as a factor is 1/p_i of the density of squarefree numbers not having p_i as a factor.
E.g., the density of even squarefree numbers is 1/p_i = 1/2 of the density of odd squarefree numbers (which means that 1/(p_i + 1) = 1/3 of the squarefree numbers are even and p_i/(p_i + 1) = 2/3 are odd) and as a consequence the n-th even squarefree number is very nearly p_i = 2 times the n-th odd squarefree number (which means that the n-th even squarefree number is very nearly (p_i + 1) = 3 times the n-th squarefree number while the n-th odd squarefree number is very nearly (p_i + 1)/ p_i = 3/2 the n-th squarefree number).
(End)
Apart from first term, these are the tau2-atoms as defined in [Anderson, Frazier] and [Lanterman]. - Michel Marcus, May 15 2019
|
|
REFERENCES
|
Richard A. Mollin, Quadratics, CRC Press, 1996, Tables B1-B3.
|
|
LINKS
|
|
|
FORMULA
|
|
|
MAPLE
|
select(numtheory:-issqrfree, [seq(i, i=2..1000, 4)]); # Robert Israel, Dec 23 2015
|
|
MATHEMATICA
|
Select[Range[2, 270, 2], SquareFreeQ] (* Harvey P. Dale, Jul 23 2011 *)
|
|
PROG
|
(Magma) [n: n in [2..262 by 2] | IsSquarefree(n)]; // Bruno Berselli, Mar 03 2011
(Haskell)
a039956 n = a039956_list !! (n-1)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,nice,easy,changed
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|