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A336745 Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005). 3

%I #20 Aug 05 2020 19:29:32

%S 1,2,6,8,9,12,18,24,28,32,36,40,54,72,80,84,96,108,117,120,128,135,

%T 144,162,196,200,216,224,234,240,243,252,270,288,324,360,384,400,405,

%U 448,468,486,496,512,540,576,588,600,625,640,648,672,675,720,756,768,775,810,819

%N Numbers m that divide the product phi(m) * sigma(m) * tau(m), where phi is the Euler totient function (A000010), sigma is the sum of divisors function (A000203) and tau is the number of divisors function (A000005).

%C If s and t are terms with gcd(s, t) = 1, then s*t is another term as phi, sigma and tau are multiplicative functions.

%C The only prime term is 2 because prime p must divide 2*(p-1)*(p+1) to be a term.

%H David A. Corneth, <a href="/A336745/b336745.txt">Table of n, a(n) for n = 1..10000</a>

%e For 24, phi(24) = 8, sigma(24) = 60 and tau(24) = 8, then 8*60*8 / 24 = 160, hence 24 is a term.

%p with(numtheory):

%p filter:= m -> irem(tau(m)*phi(m)*sigma(m), m) =0:

%p select(filter,[$1..850]);

%t Select[Range[1000], Divisible[Times @@ DivisorSigma[{0, 1}, #] * EulerPhi[#], #] &] (* _Amiram Eldar_, Aug 02 2020 *)

%o (PARI) isok(m) = !(eulerphi(m)*sigma(m)*numdiv(m) % m); \\ _Michel Marcus_, Aug 05 2020

%Y Cf. A000005, A000010, A000203, A062355.

%Y Subsequences: A000396 (perfect numbers), A005820 (tri-perfect), A027687 (4-perfect), A046060 (5-multiperfect), A046061 (6-multiperfect), A007691 (multiply-perfect numbers), A336715 (m divides phi(m)*tau(m)), A004171, A005010.

%K nonn

%O 1,2

%A _Bernard Schott_, Aug 02 2020

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Last modified March 28 20:05 EDT 2024. Contains 371254 sequences. (Running on oeis4.)