This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A173617 #28 Feb 08 2025 04:52:13
%S A173617 1,4,8,32,36,192,288,768,972,1458,5120,13122,326592,19531250,22588608,
%T A173617 46137344,171532242,110000000000,132799613957120,1618481116086272,
%U A173617 506590324238281250,8358680908399640576,162805498773679522226642,198263834416799184651812864,7852841179377049820122874642432,4299870835974154129876817272635392
%N A173617 Numbers k such that phi(tau(k)) = rad(k).
%C A173617 rad(n) is the product of the primes dividing n (A007947), tau(n) is the number of divisors of n (A000005), and phi(n) is Euler totient function (A000010).
%C A173617 Numbers k such that A163109(k) = A007947(k).
%C A173617 a(18) > 10^10. - _Donovan Johnson_, Jul 27 2011
%C A173617 From _Amiram Eldar_, Feb 08 2025: (Start)
%C A173617 1 is the only odd term in this sequence.
%C A173617 The number of terms with any given number of divisors is finite.
%C A173617 There are no terms whose number of divisors d equals 2 or in A049195, or when omega(phi(d)) > bigomega(d), where omega = A001221 and bigomega = A001222.
%C A173617 If p is a Sophie Germain prime (A005384), then 2*p^(2*p) is a term. (End)
%D A173617 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 A173617 Amiram Eldar, <a href="/A173617/a173617.txt">Mathematica code for A173617</a>, 2025.
%H A173617 P. A. MacMahon, <a href="https://doi.org/10.1112/plms/s2-19.1.75">Divisors of numbers and their continuations in the theory of partitions</a>, Proc. London Math. Soc., 19 (1919), 75-113.
%H A173617 Wacław Sierpiński, <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon42/mon4204.pdf">Number Of Divisors And Their Sum</a>, Elementary theory of numbers, Warszawa, 1964.
%H A173617 Wikipedia, <a href="http://en.wikipedia.org/wiki/Euler's_phi_function">Euler's totient function</a>.
%e A173617 8 is a term since tau(8) = 4, phi(4) = 2 and rad(8) = 2.
%e A173617 13122 is a term tau(13122) = 18, phi(18) = 6 and rad(13122) = 6.
%p A173617 with(numtheory):for n from 1 to 1000000 do :t1:= ifactors(n)[2] : t2 :=mul(t1[i][1], i=1..nops(t1)): if phi(tau(n)) = t2 then print (n): else fi : od :
%t A173617 (* First program: see the links section. *)
%t A173617 (* Second program: *)
%t A173617 q[k_] := k == 1 || EvenQ[k] && Module[{f = FactorInteger[k]}, EulerPhi[Times @@ (f[[;;,2]] + 1)] == Times @@ f[[;;, 1]]]; Select[Range[400000], q] (* _Amiram Eldar_, Feb 08 2025 *)
%o A173617 (PARI) isok(k) = if(k == 1, 1, if(k % 2, 0, my(f=factor(k)); eulerphi(numdiv(f)) == vecprod(f[,1]))); \\ _Amiram Eldar_, Feb 08 2025
%Y A173617 Cf. A000005 (tau), A000010 (phi), A007947 (rad).
%Y A173617 Cf. A001221, A001222, A001414, A005384, A049195, A062069, A163109, A173320.
%K A173617 nonn
%O A173617 1,2
%A A173617 _Michel Lagneau_, Feb 22 2010
%E A173617 a(14)-a(17) from _Donovan Johnson_, Jul 27 2011
%E A173617 a(18)-a(26) from _Amiram Eldar_, Feb 08 2025