login
This site is supported by donations to The OEIS Foundation.

 

Logo

Annual Appeal: Please make a donation (tax deductible in USA) to keep the OEIS running. Over 5000 articles have referenced us, often saying "we discovered this result with the help of the OEIS".

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A082903 a(n) = gcd(2^n, sigma_1(n)) = gcd(A000079(n), A000203(n)) also a(n) = gcd(2^n, sigma_3(n)) = gcd(A000079(n), A001158(n)). 1

%I

%S 1,1,4,1,2,4,8,1,1,2,4,4,2,8,8,1,2,1,4,2,32,4,8,4,1,2,8,8,2,8,32,1,16,

%T 2,16,1,2,4,8,2,2,32,4,4,2,8,16,4,1,1,8,2,2,8,8,8,16,2,4,8,2,32,8,1,4,

%U 16,4,2,32,16,8,1,2,2,4,4,32,8,16,2,1,2,4,32,4,4,8,4,2,2,16,8,128,16,8,4,2

%N a(n) = gcd(2^n, sigma_1(n)) = gcd(A000079(n), A000203(n)) also a(n) = gcd(2^n, sigma_3(n)) = gcd(A000079(n), A001158(n)).

%C a(n) = gcd(2^n, sigma_k(n)) when k is an odd positive integer. Proof: It suffices to show that v_2(sigma_k(n)) does not depend on k, where v_2(n) is the 2-adic valuation of n. Since v_2(ab) = v_2(a)+v_2(b) and sigma_k(n) is an arithmetic function, we need only prove it for n=p^e with p prime. If p is 2 or e is even, sigma_k(p^e) is odd, so we can disregard those cases. Otherwise, we sum the geometric series to obtain v_2(sigma_k(p^e)) = v_2(p^(k(e+1))-1)-v_2(p-1). Applying the well-known LTE Lemma (see Hossein link) Case 4 arrives at v_2(p^(k(e+1))-1)-v_2(p-1) = v_2(p+1)+v_2(k(e+1))-1. But v_2(k(e+1)) = v_2(k)+v_2(e+1), and k is odd, so we conclude that v_2(sigma_k(p^e)) = v_2(p+1)+v_2(e+1)-1, a result independent of k. - _Rafay A. Ashary_, Oct 15 2016

%H Robert Israel, <a href="/A082903/b082903.txt">Table of n, a(n) for n = 1..10000</a>

%H Amir Hossein, <a href="http://www.artofproblemsolving.com/community/c6h393335">Lifting The Exponent Lemma (Containing PDF file)</a>

%p seq(2^min(n, padic:-ordp(numtheory:-sigma(n),2)), n=1..100); # _Robert Israel_, Oct 23 2016

%o (PARI) a(n) = gcd(2^n, sigma(n)); \\ _Michel Marcus_, Oct 15 2016

%Y Cf. A000203, A001157, A001158, A082902.

%K nonn,mult

%O 1,3

%A _Labos Elemer_, Apr 22 2003

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified December 7 23:55 EST 2016. Contains 278902 sequences.