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A334337 Least positive integer m such that phi(m*n) is a cube, where phi is Euler's totient function (A000010).

%I A334337 #20 Apr 24 2020 01:05:04
%S A334337 1,1,5,4,3,4,37,2,37,2,101,2,19,37,1,1,5,36,13,1,19,101,13333,1,55,19,
%T A334337 13,19,985,1,1057,4,401,4,73,18,7,13,9,4,275,18,2649,401,9,13333,
%U A334337 169285,4,1813,50,4,73,3385,12,25,73,7,788,40371,4,3737,1057,12,2,37,401,4357,2,6537,73,5401,9,35,7,25,7,3737,9,48673,2
%N A334337 Least positive integer m such that phi(m*n) is a cube, where phi is Euler's totient function (A000010).
%C A334337 Conjecture: a(n) exists for any n > 0. In other words, for any positive integer n, there is a positive integer m with phi(m*n) equal to a cube.
%C A334337 We note that there is no positive integer m <= 10^8 with phi(107*m) equal to a fourth power.
%C A334337 The author's conjecture in A259915 implies that for any positive integer n there is a positive integer m with phi(m*n) equal to a square.
%C A334337 See also A334339 for a similar conjecture.
%H A334337 Zhi-Wei Sun, <a href="/A334337/b334337.txt">Table of n, a(n) for n = 1..1000</a>
%H A334337 P. Pollack and C. Pomerance, <a href="https://doi.org/10.1112/blms/bdt097">Square values of Euler's function</a>, Bull. London Math. Soc. 46 (2014), 403-414. <a href="https://math.dartmouth.edu/~carlp/squaretotients10.pdf">Alternative link</a>.
%H A334337 Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. See also <a href="http://arxiv.org/abs/1211.1588">arXiv</a>, arXiv:1211.1588 [math.NT], 2012-2017. (Cf. Conjecture 4.5.)
%e A334337 a(3) = 5 with phi(3*5) = 2^3.
%e A334337 a(7) = 37 with phi(7*37) = 6^3.
%e A334337 a(863) = 21176773 with phi(863*21176773) = 17293606056 = 2586^3.
%t A334337 CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)];
%t A334337 phi[n_]:=phi[n]=EulerPhi[n];
%t A334337 tab={};Do[m=0;Label[aa];m=m+1;If[CQ[phi[m*n]],tab=Append[tab,m],Goto[aa]],{n,1,80}];tab
%o A334337 (PARI) a(n) = my(m=1); while (!ispower(eulerphi(n*m), 3), m++); m; \\ _Michel Marcus_, Apr 23 2020
%Y A334337 Cf. A000010, A000578, A039771,  A259915, A259916, A280988, A334339.
%K A334337 nonn
%O A334337 1,3
%A A334337 _Zhi-Wei Sun_, Apr 23 2020