A030165 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)*...

36, 252, 288, 756, 1116, 1512, 2016, 4572, 6048, 8928, 24192, 36576, 62208, 115200, 136080, 294876, 806400, 2359008, 2419200, 3571200, 4147200, 4718556, 6193152, 10782720, 14630400, 18874332, 20575296, 29030400, 30108672, 37748448, 58786560

Offset

1,1

Comments

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

Links

Donovan Johnson, Table of n, a(n) for n = 1..45 (terms < 10^11)

Mathematica

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 *)

Crossrefs

Cf. A047994, A030164.

Sequence in context: A074363 A219888 A233363 * A282099 A247840 A017342

Adjacent sequences: A030162 A030163 A030164 * A030166 A030167 A030168

Keyword

nonn

Author

Yasutoshi Kohmoto

Extensions

Corrected and extended by Farideh Firoozbakht, Jul 08 2006

Status

approved