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A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.

%I A334339 #24 Nov 05 2020 19:45:10
%S A334339 1,51,34,291,22,17,1,1347,597,11,10,97,892,51,46,1758,6,3540,343,1649,
%T A334339 34,5,30,449,2928,446,199,291,472,23,34,879,235,3,22,1770,8661,356,
%U A334339 3007,1593,884,17,241,298,1416,15,22,586,133,1464,2,223,3,1180,2,1347,711,236,232,1062,1200,17,597,96771,586,265,577,485,10,11
%N A334339 Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.
%C A334339 Conjecture: a(n) exists for any n > 0. In other words, for any positive integer n, there is a positive integer m with sigma(m * n) equal to a cube.
%C A334339 The author's conjecture in A259915 implies that for each n = 1, 2, 3, ... there is a positive integer m with sigma(m * n) equal to a square.
%C A334339 See also A334337 for a similar conjecture.
%H A334339 Zhi-Wei Sun, <a href="/A334339/b334339.txt">Table of n, a(n) for n = 1..1000</a>
%H A334339 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 A334339 a(2) = 51 with sigma(2*51) = 216 = 6^3.
%e A334339 a(4) = 291 with sigma(4*291) = 2744 = 14^3.
%e A334339 a(578) = 34312749 with sigma(578*34312749) = 42144192000 = 3480^3.
%e A334339 a(673) = 49061802 with sigma(673*49061802) = 66135317184 = 4044^3.
%t A334339 cubeQ[n_] := cubeQ[n] = IntegerQ[n^(1/3)];
%t A334339 sigma[n_] := sigma[n] = DivisorSigma[1, n];
%t A334339 tab = {}; Do[m = 0; Label[aa]; m = m + 1; If[cubeQ[sigma[m * n]], tab = Append[tab, m], Goto[aa]], {n, 70}]; tab
%t A334339 lpi[n_]:=Module[{k=1},While[!IntegerQ[Surd[DivisorSigma[1,n*k],3]],k++]; k]; Array[lpi,70] (* _Harvey P. Dale_, Nov 05 2020 *)
%o A334339 (PARI) a(n) = my(m=1); while (!ispower(sigma(n*m), 3), m++); m; \\ _Michel Marcus_, Apr 23 2020
%Y A334339 Cf. A000203, A000578, A020477, A259915, A334337.
%K A334339 nonn
%O A334339 1,2
%A A334339 _Zhi-Wei Sun_, Apr 23 2020
%E A334339 Corrected and extended by _Harvey P. Dale_, Nov 05 2020