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A114076 Numbers k such that k * phi(k) is a cube.

%I A114076 #40 Jun 04 2024 16:41:51
%S A114076 1,4,32,50,72,225,256,400,576,900,1944,2048,2166,2312,2646,3200,4107,
%T A114076 4563,4608,5202,6075,6250,7200,7225,15125,15552,16384,16428,17328,
%U A114076 18252,18496,21168,23762,24300,25600,28125,28900,35378,36864,41616,50000,52488,57600
%N A114076 Numbers k such that k * phi(k) is a cube.
%C A114076 From _Robert Israel_, Sep 06 2020: (Start)
%C A114076 If n > 1 is in the sequence, A071178(n) == 2 (mod 3).
%C A114076 If p=2^(2^k)+1 is in A019434, includes 2^a*p^b where a == 2^k-1 (mod 3) and b == 2 (mod 3).
%C A114076 If members m and n are coprime, then m*n is in the sequence.
%C A114076 If n is in the sequence and prime p divides n, then p^3*n is in the sequence. (End)
%C A114076 To look for terms it suffices to see if cubes have a divisors pair (k, m) such that phi(m) = k. - _David A. Corneth_, May 21 2024
%H A114076 David A. Corneth, <a href="/A114076/b114076.txt">Table of n, a(n) for n = 1..8565</a> (first 300 terms from Robert Israel, terms <= 5*10^11)
%H A114076 David A. Corneth, <a href="/A114076/a114076.gp.txt">PARI program</a>
%e A114076 phi(1944) * 1944 = 1259712 = 108^3.
%p A114076 filter:= proc(n) local F;
%p A114076   F:= ifactors(n*numtheory:-phi(n))[2];
%p A114076   type(map(t -> t[2]/3, F), list(integer));
%p A114076 end proc:
%p A114076 select(filter, [$1..10^5]); # _Robert Israel_, Sep 06 2020
%t A114076 Select[Range[57600],IntegerQ[(# EulerPhi[#])^(1/3)]&] (* _Stefano Spezia_, May 29 2024 *)
%o A114076 (PARI) isok(n) = ispower(n*eulerphi(n), 3); \\ _Michel Marcus_, Jan 22 2014
%o A114076 (PARI) upto(n)= res = List(); forfactored(i = 1, n, if(ispower(i[1] * eulerphi(i[2]), 3), listput(res, i[1]); ) ); res  \\ _David A. Corneth_, Dec 08 2022
%o A114076 (PARI) \\ See Corneth link
%o A114076 (Python)
%o A114076 from sympy import integer_nthroot, totient as phi
%o A114076 def ok(k): return integer_nthroot(k * phi(k), 3)[1]
%o A114076 print([k for k in range(1, 60000) if ok(k)]) # _Michael S. Branicky_, Dec 08 2022
%Y A114076 Aside from the first term, a subsequence of A070003. A013731 is a subsequence.
%Y A114076 Cf. A000010, A071178, A019434.
%K A114076 nonn
%O A114076 1,2
%A A114076 _Giovanni Resta_, Feb 13 2006
%E A114076 More terms from _Michel Marcus_, Jan 22 2014