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Numbers k such that d(k) (number of divisors) divides phi(k) (Euler function) divides sigma(k) (sum of divisors).
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%I #20 Apr 13 2024 05:19:36

%S 1,3,15,30,35,56,70,78,105,140,168,190,210,248,264,357,420,570,616,

%T 630,714,744,812,840,910,1045,1240,1485,1672,1848,2090,2214,2436,2580,

%U 2730,3080,3135,3339,3596,3720,3956,4064,4180,4522,4674,5016,5049,5278,5396

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

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

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

%H Donovan Johnson, <a href="/A020493/b020493.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%o (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

%Y Cf. A000005, A000010, A000203.

%Y Intersection of A020491 and A020492.

%K nonn

%O 1,2

%A _David W. Wilson_

%E Wells incorrectly has 52 instead of 56.