A279894 Numbers m which can be written as x*y with phi(x)*sigma(y) = m, 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, 12, 18, 24, 48, 54, 56, 84, 96, 112, 120, 162, 168, 192, 224, 240, 252, 336, 360, 384, 448, 468, 480, 486, 588, 600, 672, 720, 756, 768, 896, 936, 960, 992, 1080, 1176, 1200, 1344, 1440, 1458, 1536, 1764, 1792, 1800, 1872, 1920, 1984, 2160, 2268

Offset

1,2

Comments

As phi(x) is even for any integer x > 1, only the first term 1 is odd.

If n is a perfect number, then 2*n = phi(2)*sigma(n) is a term of the sequence.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..1000

Example

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

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]]==m, n=n+1; Print[n, " ", m]; Goto[aa]], {i, 1, Ld[m]}]; Label[aa]; Continue, {m, 1, 2300}]

Crossrefs

Cf. A000010, A000203, A000396.

Sequence in context: A359957 A031478 A073455 * A209309 A020333 A044831

Adjacent sequences: A279891 A279892 A279893 * A279895 A279896 A279897

Keyword

nonn

Author

Zhi-Wei Sun, Dec 22 2016

Status

approved