A030165 - OEIS

%I #19 Jun 27 2022 23:35:36

%S 36,252,288,756,1116,1512,2016,4572,6048,8928,24192,36576,62208,

%T 115200,136080,294876,806400,2359008,2419200,3571200,4147200,4718556,

%U 6193152,10782720,14630400,18874332,20575296,29030400,30108672,37748448,58786560

%N Numbers m such that uphi(sigma(m)) = 2m, where the unitary phi function (A047994) is defined by: if x = p1^r1*p2^r2*p3^r3*... then uphi(x) = (p1^r1 - 1)*(p2^r2 - 1)*(p3^r3 - 1)*...

%C Theorem: If m is in the sequence, sigma(m) is an odd number, 2^n-1 is a prime greater than 3 (a Mersenne prime) and gcd(m, 2^n-1)=1, then m*(2^n-1) is in the sequence (the proof is easy). One of the results of this theorem is: If p=2^n-1 is a prime greater than 3 then 36*p, 288*p, 115200*p and 4147200*p are in the sequence. - _Farideh Firoozbakht_, Jul 08 2006

%H Donovan Johnson, <a href="/A030165/b030165.txt">Table of n, a(n) for n = 1..45</a> (terms < 10^11)

%t uphi[n_] := (A = FactorInteger[n]; l = Length[A]; Product[A[[k]][[1]] ^A[[k]][[2]] - 1, {k, l}]); Do[If[uphi[DivisorSigma[1, n]] == 2n, Print[n]], {n, 70000000}] (* _Farideh Firoozbakht_, Jul 08 2006 *)

%Y Cf. A047994, A030164.

%K nonn

%O 1,1

%A _Yasutoshi Kohmoto_

%E Corrected and extended by _Farideh Firoozbakht_, Jul 08 2006