A015759 Numbers k such that phi(k) | sigma_2(k).

1, 2, 3, 6, 22, 33, 66, 750, 27798250, 41697375, 76745867, 83394750, 153491734, 207656250, 230237601, 460475202, 917342250, 969062500, 2907187500, 4528006153, 5952812500, 9056012306, 13584018459, 17858437500, 27168036918, 31979062500, 57559400250

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn

Comments

sigma_2(k) is the sum of the squares of the divisors of k (A001157).

All of these terms are solutions to relations for all j as follows: {sigma(j,x)/phi(x) is an integer for exponents j=4k+2}. Proof is possible by individual managements in the knowledge of divisors of x and phi(x). Compare with A015765, A015768, etc. - Labos Elemer, May 25 2004

Crossrefs

Cf. A000010, A001157, A093643.

Cf. A015765, A015768, A094470.

Programs

Mathematica

Do[ If[ IntegerQ[ DivisorSigma[2, n]/EulerPhi[n]], Print[n]], {n, 1, 10^7}]
Empirical test for very high power sums of divisors [e.g., d^2802]. Table[{4*j+2, Union[Table[IntegerQ[DivisorSigma[4*j+2, Part[t, k]]/EulerPhi[Part[t, k]]], {k, 1, 13}]]}, {j, 0, 700}] Output = {True} for all 4j+2. Here t=A015759. (* Labos Elemer, May 20 2004 *)

Extensions

a(9)-a(13) from Labos Elemer, May 20 2004

a(14)-a(18) from Donovan Johnson, Feb 05 2010

a(19)-a(27) from Donovan Johnson, Jun 18 2011