A328651 Composite k for which lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)).

135, 546, 672, 9585, 24570, 51510, 63855, 190008, 251370, 323730, 372438, 486180, 510570, 723550, 819000, 1058910, 1282365, 1284192, 1356030, 3506390, 5416200, 5604480, 6298625, 15593760, 17813250, 18009000, 20740590, 26759370, 27027000, 27081000, 29795040

Offset

1,1

Comments

Composite numbers k verifying equation A009230(k) + A009262(k) = A009242(k).

For any prime number p >= 3 the equality lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)) is satisfied.

The sequence terms are the composite numbers for which the equality is true.

Links

Table of n, a(n) for n=1..31.

Example

For k = 135 = 3^3 * 5, tau(k) = 4 * 2 = 2^3, phi(k) = 2 * 3^2 * 4 = 2^3 * 3^2 , sigma(k) = 2^4 * 3 * 5, lcm(k, tau(k)) + lcm(k, phi(k)) =  2^3 * 3^3 * 5 + 2^3 * 3^3 * 5 = 2^4 * 3^3 * 5 and lcm(k, sigma(k)) = lcm(3^3 * 5, 2^4 * 3 * 5) = 2^4 * 3^3 * 5.

Mathematica

aQ[n_] := CompositeQ[n] && LCM[n, EulerPhi[n]] + LCM[n, DivisorSigma[0, n]] == LCM[n, DivisorSigma[1, n]]; Select[Range[3*10^6], aQ] (* Amiram Eldar, Oct 23 2019 *)

Prog

(Magma) [k: k in [1..6000000]| not IsPrime(k) and Lcm(k, NumberOfDivisors(k))+Lcm(k, EulerPhi(k)) eq Lcm(k, SumOfDivisors(k))];
(PARI) isok(k) = !isprime(k) && (lcm(k, numdiv(k)) + lcm(k, eulerphi(k)) == lcm(k, sigma(k))); \\ Michel Marcus, Oct 24 2019

Crossrefs

Cf. A000005, A000010, A000203, A009242, A009230, A009262, A326416.

Sequence in context: A335328 A215173 A225360 * A159201 A362400 A211680

Adjacent sequences: A328648 A328649 A328650 * A328652 A328653 A328654

Keyword

nonn

Author

Marius A. Burtea, Oct 23 2019

Status

approved