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

1,2

Comments

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

References

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

Links

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A000005, A000010, A000203.

Intersection of A020491 and A020492.

Sequence in context: A202506 A053519 A039666 * A087183 A297851 A298088

Adjacent sequences: A020490 A020491 A020492 * A020494 A020495 A020496

Keyword

nonn

Author

David W. Wilson

Extensions

Wells incorrectly has 52 instead of 56.

Status

approved