A173327 Numbers k such that tau(phi(k))= sopf(k).

4, 45, 48, 75, 160, 180, 252, 294, 300, 315, 336, 351, 378, 396, 475, 507, 560, 605, 616, 650, 833, 936, 1216, 1375, 1452, 1690, 1805, 1920, 2023, 2112, 2200, 2349, 2496, 2736, 3211, 3520, 3648, 4095, 4160, 4256, 4332, 4389, 4464, 4477, 4508, 4620, 4693

Offset: 1

Author

Michel Lagneau, Feb 16 2010

Keywords

• nonn

Comments

tau(k) is the number of divisors of k (A000005); phi(k) is the Euler totient function (A000010); and sopf(k) is the sum of the distinct primes dividing k without repetition (A008472).

Examples

			4 is in the sequence because phi(4) = 2, tau(2)=2 and sopf(4)=2 ;
45 is in the sequence because phi(45) = 24, tau(24)=8 and sopf(45)=8.
		

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
• A. Bogomolny, Euler Function and Theorem
• 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.
• Wikipedia, Euler's totient function

Crossrefs

See A173326, A062069. Cf. A001414, A008472(sopfr), A001222, A062821.

Programs

Magma

[ m:m in [2..5100]|#Divisors(EulerPhi(m)) eq &+PrimeDivisors(m)]; // Marius A. Burtea, Jul 10 2019

Maple

for n from 1 to 150000 do : t1:= ifactors(n)[2] : t2 :=sum(t1[i][1], i=1..nops(t1)):if tau(phi(n)) = t2 then print (n): else fi : od :

Mathematica

tpsQ[n_]:=DivisorSigma[0,EulerPhi[n]]==Total[Transpose[FactorInteger[n]][[1]]]; Select[Range[5000],tpsQ] (* Harvey P. Dale, Apr 10 2013 *)

PARI

sopf(n)=my(f=factor(n)[1,]);sum(i=1,#f,f[i])
is(n)=numdiv(eulerphi(n))==sopf(n) \\ Charles R Greathouse IV, May 20 2013

Formula

Numbers n such that A062821(n)= A008472(n)

Extensions

Added punctuation to the examples. Corrected and edited by Michel Lagneau, Apr 25 2010

Edited by D. S. McNeil, Nov 20 2010