|
%I #15 Jul 09 2019 10:58:33
%S 1,4,36,54,96,200,448,1280,2700,4500,5103,9720,11264,14112,14580,
%T 17280,26624,32928,48000,54432,71442,75000,81648,152064,184320,187500,
%U 258048,307200,350000,637875,1250235,1344560,1557504,2044416,2187500,2367488,3234816
%N Numbers k such that tau(phi(k)) = rad(k).
%C 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).
%D 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.
%H Amiram Eldar, <a href="/A173618/b173618.txt">Table of n, a(n) for n = 1..100</a>
%H W. Sierpinski, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler's_phi_function">Euler's totient function</a>
%F k such that A062821(k) = A007947(k).
%e phi(4) = 2, tau(2) = 2 and rad(4) = 2 phi(36) = 12, tau(12) = 6 and rad(36) = 6
%p 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 :
%t rad[n_] := Times @@ (First@# & /@ FactorInteger[n]); Select[Range[10^5], DivisorSigma[0, EulerPhi[#]] == rad[#] &] (* _Amiram Eldar_, Jul 09 2019*)
%o (PARI) isok(k) = numdiv(eulerphi(k)) == factorback(factorint(k)[, 1]); \\ _Michel Marcus_, Jul 09 2019
%Y Cf. A000005, A000010, A062069, A062821, A007947, A173326.
%K nonn
%O 1,2
%A _Michel Lagneau_, Feb 22 2010
%E a(30)-a(37) from _Donovan Johnson_, Jul 27 2011
|
