A306655 Numbers n such that lcm(sigma(n), n) = tau(n) * sigma(n) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

1, 2, 18, 468, 9360, 10880, 79360, 84480, 387072, 777216, 3801600, 7282688, 15037440, 17418240, 27067392, 65544192, 752903424, 1218032640, 4227842304, 4737761280, 6410638080, 11949932544, 19327057920, 26372530800, 37645171200, 224956569600, 243520929792, 876611248128

Offset: 1

Author

Jaroslav Krizek, Mar 03 2019

Keywords

• nonn

Comments

Numbers n such that A009242(n) = A000005(n) * A000203(n) = A064840(n).

Also numbers n such that A017666(n) = denominator(sigma(n)/n) = tau(n) = A000005(n).

a(29) > 10^12. - Giovanni Resta, Mar 04 2019

Examples

			18 is a term because lcm(sigma(18), 18) = lcm(39, 18) = 234 = tau(18) * sigma(18) = 6 * 39.
		

Crossrefs

Cf. A000005, A000203, A009242, A064840.

Cf. A069810 (gcd(sigma(n), n) = tau(n)).

Programs

Magma

[n: n in [1..1000000] | LCM(SumOfDivisors(n), n) eq NumberOfDivisors(n)* SumOfDivisors(n)]

Mathematica

Select[Range[1000000], LCM[DivisorSigma[1, #], #] == DivisorSigma[0, #] * DivisorSigma[1, #]&] (* Vaclav Kotesovec, Mar 04 2019 *)

PARI

isok(n) = my(sn = sigma(n)); lcm(sn, n) == sn*numdiv(n); \\ Michel Marcus, Mar 04 2019

Extensions

a(13)-a(16) from Vaclav Kotesovec, Mar 04 2019

a(17) from Daniel Suteu, Mar 04 2019

a(18)-a(28) from Giovanni Resta, Mar 04 2019