

A214682


Remove 2s that do not contribute to a factor of 4 from the prime factorization of n.


4



1, 1, 3, 4, 5, 3, 7, 4, 9, 5, 11, 12, 13, 7, 15, 16, 17, 9, 19, 20, 21, 11, 23, 12, 25, 13, 27, 28, 29, 15, 31, 16, 33, 17, 35, 36, 37, 19, 39, 20, 41, 21, 43, 44, 45, 23, 47, 48, 49, 25, 51, 52, 53, 27, 55, 28, 57, 29
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,3


COMMENTS

In this sequence, the number 4 exhibits some characteristics of a prime number since all extraneous 2's have been removed from the prime factorizations of all other numbers.


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000


FORMULA

a(n)=(n*4^(v_4(n)))/(2^(v_2(n))) where v_k(n) is the kadic valuation of n. That is, v_k(n) is the largest power of k, a, such that k^a divides n.


EXAMPLE

For n=8, v_4(8)=1, v_2(8)=3, so a(8)=(8*4^1)/(2^3)=4.
For n=12, v_4(12)=1, v_2(12)=2, so a(12)=(12*4^1)/(2^2)=12.
For n odd, a(n)=n since n has no factors of 2 (or 4).


PROG

(Sage)
C=[]
for i in [1..n]:
...C.append(i*(4^(Integer(i).valuation(4))/(2^(Integer(i).valuation(2)))
(PARI) a(n)=n>>(valuation(n, 2)%2) \\ Charles R Greathouse IV, Jul 26 2012


CROSSREFS

Cf. A214681, A214685.
Sequence in context: A270027 A271726 A123901 * A093395 A176774 A126352
Adjacent sequences: A214679 A214680 A214681 * A214683 A214684 A214685


KEYWORD

easy,nonn,mult


AUTHOR

Tyler Ball, Jul 25 2012


STATUS

approved



