A173328 Numbers k such that phi(tau(k)) = tau(sopf(k)).

4, 6, 8, 9, 10, 12, 18, 20, 22, 25, 27, 30, 32, 34, 44, 49, 50, 58, 60, 68, 70, 82, 90, 102, 104, 105, 116, 118, 121, 125, 135, 140, 142, 150, 152, 164, 169, 174, 182, 189, 190, 195, 202, 204, 208, 214, 231, 236, 238, 242, 243, 246, 248, 252, 274, 284, 285, 286

Offset: 1

Author

Michel Lagneau, Feb 16 2010

Keywords

• nonn

Comments

Sopf(n) = A008472(n) is the sum of the distinct primes dividing n, tau(n) = A000005(n) is the number of divisors of n, phi = A000010 is Euler's totient function.

Examples

			4 is in the sequence because tau(4) = 3, phi(3) = 2, sopf(4) = 2 and tau(2) = 2.
6 is in the sequence because tau(6) = 4, phi(6) = 2, sopf(6) = 5 and tau(5) = 2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1919), 75-113.
• Wacław Sierpiński, Number Of Divisors And Their Sum, Elementary theory of numbers, Warszawa, 1964.

Crossrefs

Cf. A000005 (tau), A000010 (phi), A008472 (sopfr), A163109.

Programs

Maple

isA173328 := proc(n)
        numtheory[phi](numtheory[tau](n)) = numtheory[tau](A008472(n)) ;
end proc:
for n from 1 to 300 do
        if isA173328(n) then
                printf("%d,",n);
        end if;
end do: # R. J. Mathar, Nov 07 2011

Mathematica

Select[Range[2,300],EulerPhi[DivisorSigma[0,#]]==DivisorSigma[0, Total[ FactorInteger[#][[All,1]]]]&] (* Harvey P. Dale, May 30 2017 *)

PARI

isok(k) = if(k == 1, 0, my(f=factor(k)); eulerphi(numdiv(f)) == numdiv(vecsum(f[,1]))); \\ Amiram Eldar, Feb 08 2025

Formula

{k : A163109(k) = tau(A008472(k))}.