A173334 Numbers k such that tau(phi(k)) = phi(sum-of-prime-divisors(k)).

2, 3, 15, 18, 24, 28, 30, 33, 39, 50, 52, 55, 80, 132, 133, 152, 169, 186, 187, 190, 195, 207, 215, 217, 222, 230, 238, 247, 261, 266, 305, 319, 333, 340, 352, 369, 371, 414, 481, 484, 494, 496, 497, 506, 516, 522, 559, 574, 580, 611, 644, 646, 660, 671, 689

Offset: 1

Author

Michel Lagneau, Feb 16 2010

Keywords

• nonn

Comments

Numbers k such that A000005(A000010(k)) = A000010(A008472(k)).

Examples

			For n=15, tau(phi(15)) = tau(8)=4 equals phi(A008472(15))=phi(8) = 4, which adds 15 to the sequence.
For n=18, tau(phi(18)) = tau(6) =4 equals phi(A008472(18)) = phi(5) = 4, which adds 18 to the sequence.
		

References

• M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 840.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• P. A. MacMahon, Divisors of numbers and their continuations in the theory of partitions, Proc. London Math. Soc., 19 (1921), 75-113.
• W. Sierpinski, Number Of Divisors And Their Sum, Monogr. Matemat. 42 (1964) chapter IV
• Wikipedia, Euler's totient function

Crossrefs

Cf. A000005, A000010, A008472, A062821.

Programs

Magma

[m:m in [2..700]|#Divisors(EulerPhi(m)) eq EulerPhi(&+PrimeDivisors(m))]; // Marius A. Burtea, Jul 10 2019

Maple

with(numtheory): for n from 1 to 1800 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = phi(t2) then print (n): else fi : od :

Mathematica

Select[Range[2, 700], DivisorSigma[0, EulerPhi[#]] == EulerPhi[Total[FactorInteger[#][[All, 1]]]] &]
(* Jean-François Alcover, May 19 2011 *)

PARI

isok(n) = numdiv(eulerphi(n)) == eulerphi(vecsum(factor(n)[, 1])); \\ Michel Marcus, Jul 10 2019

Formula

{n : A062821(n)= phi(A008472(n))}.

Extensions

Removed sopf acronym. Updated references and links - R. J. Mathar, Mar 10 2010