A020493 Numbers k such that d(k) (number of divisors) divides phi(k) (Euler function) divides sigma(k) (sum of divisors).

1, 3, 15, 30, 35, 56, 70, 78, 105, 140, 168, 190, 210, 248, 264, 357, 420, 570, 616, 630, 714, 744, 812, 840, 910, 1045, 1240, 1485, 1672, 1848, 2090, 2214, 2436, 2580, 2730, 3080, 3135, 3339, 3596, 3720, 3956, 4064, 4180, 4522, 4674, 5016, 5049, 5278, 5396

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Numbers k such that sigma_0(k) divides phi(k) divides sigma_1(k).

Examples

			210 has 16 divisors, which divides phi(210) = 48, which in turn divides sigma(210) = 576, so 210 is a term of the sequence.
		

References

• David Wells, Curious and interesting numbers, Penguin Books, p. 130.

Links

• Donovan Johnson, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A000203.

Intersection of A020491 and A020492.

Programs

Mathematica

q[n_] := And @@ IntegerQ /@ Ratios @ {DivisorSigma[0, n], EulerPhi[n], DivisorSigma[1, n]}; Select[Range[6000], q] (* Amiram Eldar, Apr 13 2024 *)

PARI

for(n=1, 1e3, if(sigma(n)%eulerphi(n)==0, if(sigma(n)%numdiv(n)==0, if(eulerphi(n)%numdiv(n)==0, print1(n, ", "))))) \\ Felix Fröhlich, Aug 08 2014

Extensions

Wells incorrectly has 52 instead of 56.