A279915 Numbers m that can be written as x*y with phi(x)*sigma(y) = 2*x*y, where x and y are positive integers, phi(.) is Euler's totient function and sigma(y) is the sum of all positive divisors of y.

6, 28, 84, 120, 234, 360, 496, 588, 600, 1080, 1638, 2016, 3000, 3042, 3240, 3276, 4116, 4680, 6048, 7440, 8128, 9720, 11466, 14040, 15000, 18144, 22320, 22932, 23400, 28812, 29160, 30240, 32640, 32760, 37200, 39546, 42120, 42588, 54432, 55800, 60480, 60840, 65520, 66960

Offset

1,1

Comments

Conjecture: (i) All the terms are even. Moreover, if x and y are positive integers with phi(x)*sigma(y) = 2*x*y, then y must be even.

(ii) If x and y are positive integers with phi(x)*sigma(y) = 2*x*y, then x = 1 or 3 | y. Thus, any term of the sequence is either a perfect number or a multiple of three.

As phi(1) = 1, the sequence contains all perfect numbers, and part (i) of the above conjecture implies the well-known conjecture that there are no odd perfect numbers.

We consider the terms of this sequence as natural extensions of perfect numbers. There are a total of 433 terms not exceeding 10^8, and they are all even.

It is easy to see that a positive integer n with sigma(n) odd must be a square or twice a square.

See also A279894 for a similar sequence.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..433

Example

a(1) = 6 since 6 = 1*6 with phi(1)*sigma(6) = 2*6.
a(3) = 84 since 84 = 7*12 with phi(7)*sigma(12) = 2*84.

Mathematica

sigma[n_]:=sigma[n]=DivisorSigma[1, n];
phi[n_]:=phi[n]=EulerPhi[n];
Dv[m_]:=Dv[m]=Divisors[m];
Ld[m_]:=Ld[m]=Length[Dv[m]];
n=0; Do[Do[If[sigma[Part[Dv[m], i]]phi[m/Part[Dv[m], i]]==2m, n=n+1; Print[n, " ", m]; Goto[aa]], {i, 1, Ld[m]}]; Label[aa]; Continue, {m, 1, 70000}]
(* Second program *)
Select[Range[10^5], Function[n, Total@ Boole@ Map[EulerPhi[#1] DivisorSigma[1, #2] == 2 #1 #2 & @@ {#, n/#} &, Divisors@ n] > 0]] (* Michael De Vlieger, Dec 23 2016 *)

Crossrefs

Cf. A000010, A000203, A000396, A279894.

Sequence in context: A144945 A308585 A202956 * A300906 A222198 A302650

Adjacent sequences: A279912 A279913 A279914 * A279916 A279917 A279918

Keyword

nonn

Author

Zhi-Wei Sun, Dec 22 2016

Status

approved