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A334339
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Least positive integer m such that sigma(m * n) is a cube, where sigma(k) is the sum of the divisors of k.
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3
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1, 51, 34, 291, 22, 17, 1, 1347, 597, 11, 10, 97, 892, 51, 46, 1758, 6, 3540, 343, 1649, 34, 5, 30, 449, 2928, 446, 199, 291, 472, 23, 34, 879, 235, 3, 22, 1770, 8661, 356, 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
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OFFSET
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1,2
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COMMENTS
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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.
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.
See also A334337 for a similar conjecture.
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LINKS
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Zhi-Wei Sun, Conjectures on representations involving primes, 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 arXiv, arXiv:1211.1588 [math.NT], 2012-2017. (Cf. Conjecture 4.5.)
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EXAMPLE
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a(2) = 51 with sigma(2*51) = 216 = 6^3.
a(4) = 291 with sigma(4*291) = 2744 = 14^3.
a(578) = 34312749 with sigma(578*34312749) = 42144192000 = 3480^3.
a(673) = 49061802 with sigma(673*49061802) = 66135317184 = 4044^3.
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MATHEMATICA
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cubeQ[n_] := cubeQ[n] = IntegerQ[n^(1/3)];
sigma[n_] := sigma[n] = DivisorSigma[1, n];
tab = {}; Do[m = 0; Label[aa]; m = m + 1; If[cubeQ[sigma[m * n]], tab = Append[tab, m], Goto[aa]], {n, 70}]; tab
lpi[n_]:=Module[{k=1}, While[!IntegerQ[Surd[DivisorSigma[1, n*k], 3]], k++]; k]; Array[lpi, 70] (* Harvey P. Dale, Nov 05 2020 *)
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PROG
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(PARI) a(n) = my(m=1); while (!ispower(sigma(n*m), 3), m++); m; \\ Michel Marcus, Apr 23 2020
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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