A173326 -id:A173326 - cp's OEIS frontend

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

Michel Lagneau, Feb 16 2010

nonn

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).

			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.

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.

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

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

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

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 :

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

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

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

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

Edited by D. S. McNeil, Nov 20 2010

A173618 Numbers k such that tau(phi(k)) = rad(k).

Original entry on oeis.org

1, 4, 36, 54, 96, 200, 448, 1280, 2700, 4500, 5103, 9720, 11264, 14112, 14580, 17280, 26624, 32928, 48000, 54432, 71442, 75000, 81648, 152064, 184320, 187500, 258048, 307200, 350000, 637875, 1250235, 1344560, 1557504, 2044416, 2187500, 2367488, 3234816

Offset: 1

Michel Lagneau, Feb 22 2010

nonn

rad(k) is the product of the primes dividing k (A007947), tau(k) is the number of divisors of k (A000005), phi(k) is the Euler totient function (A000010).

			phi(4) = 2, tau(2) = 2 and rad(4) = 2 phi(36) = 12, tau(12) = 6 and rad(36) = 6

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.

Amiram Eldar, Table of n, a(n) for n = 1..100
W. Sierpinski, Number Of Divisors And Their Sum
Wikipedia, Euler's totient function

Cf. A000005, A000010, A062069, A062821, A007947, A173326.

with(numtheory):for n from 1 to 1000000 do : t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if tau(phi(n))= t2 then print (n): else fi: od :

rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); Select[Range[10^5], DivisorSigma[0, EulerPhi[#]] == rad[#] &] (* Amiram Eldar, Jul 09 2019*)

isok(k) = numdiv(eulerphi(k)) == factorback(factorint(k)[, 1]); \\ Michel Marcus, Jul 09 2019

k such that A062821(k) = A007947(k).

a(30)-a(37) from Donovan Johnson, Jul 27 2011

cp's OEIS Frontend

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

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