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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. 1
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 18 02:23 EDT 2019. Contains 328135 sequences. (Running on oeis4.)