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

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

Author

Michel Lagneau, Feb 22 2010

Keywords

• nonn

Comments

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).

Examples

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

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..100
• W. Sierpinski, Number Of Divisors And Their Sum
• Wikipedia, Euler's totient function

Crossrefs

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

Programs

Maple

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 :

Mathematica

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

PARI

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

Formula

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

Extensions

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