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

1,1

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).

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

Formula

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

Example

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.

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 *)

Prog

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

Crossrefs

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

Sequence in context: A327196 A024254 A167781 * A075029 A369228 A070225

Adjacent sequences: A173324 A173325 A173326 * A173328 A173329 A173330

Keyword

nonn

Author

Michel Lagneau, Feb 16 2010

Extensions

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

Edited by D. S. McNeil, Nov 20 2010

Status

approved