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A259916
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Least positive integer k such that sigma(k) and phi(k*n) are both squares, where sigma(k) is the sum of all positive divisors of k, and phi(.) is Euler's totient function.
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4
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1, 1, 210, 3, 1, 170, 81, 1, 70, 1, 400, 1, 210, 81, 357, 3, 1, 119, 3, 3, 3, 651, 1990, 170, 66, 70, 210, 884, 3810, 357, 1066, 1, 217, 1, 81, 3, 1, 3, 70, 1, 22, 3, 1624, 217, 119, 3383, 11510, 1, 364, 22, 210, 81, 8743, 170, 510, 81, 1, 1270, 2902, 1, 385, 1155, 1, 3, 357, 217, 966, 3, 4179, 81
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OFFSET
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1,3
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COMMENTS
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The conjecture in A259915 implies that a(n) exists for any n > 0.
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REFERENCES
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Zhi-Wei Sun, Problems on combinatorial properties of primes, in: M. Kaneko, S. Kanemitsu and J. Liu (eds.), Number Theory: Plowing and Starring through High Wave Forms, Proc. 7th China-Japan Seminar (Fukuoka, Oct. 28 - Nov. 1, 2013), Ser. Number Theory Appl., Vol. 11, World Sci., Singapore, 2015, pp. 169-187.
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LINKS
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EXAMPLE
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a(3) = 210 since sigma(210) = 576 =24^2 and phi(210*3) = 144 = 12^2.
a(719) = 42862647 since sigma(42862647) = 58003456 = 7616^2 and phi(42862627*719) = phi(30818243193) = 20210602896 = 142164^2.
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MATHEMATICA
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SQ[n_]:=IntegerQ[Sqrt[n]]
sigma[n_]:=DivisorSigma[1, n]
Do[k=0; Label[aa]; k=k+1; If[SQ[sigma[k]]&&SQ[EulerPhi[k*n]], Goto[bb], Goto[aa]]; Label[bb]; Print[n, " ", k]; Continue, {n, 1, 70}]
lpi[n_]:=Module[{k=1}, While[!IntegerQ[Sqrt[DivisorSigma[1, k]]]|| !IntegerQ[ Sqrt[ EulerPhi[ n*k]]], k++]; k]; Array[lpi, 70] (* Harvey P. Dale, Jul 17 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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STATUS
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approved
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