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%I #13 Jul 17 2020 15:13:03
%S 1,1,210,3,1,170,81,1,70,1,400,1,210,81,357,3,1,119,3,3,3,651,1990,
%T 170,66,70,210,884,3810,357,1066,1,217,1,81,3,1,3,70,1,22,3,1624,217,
%U 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
%N 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.
%C The conjecture in A259915 implies that a(n) exists for any n > 0.
%D 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.
%H Zhi-Wei Sun, <a href="/A259916/b259916.txt">Table of n, a(n) for n = 1..1000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641 [math.NT], 2014.
%e a(3) = 210 since sigma(210) = 576 =24^2 and phi(210*3) = 144 = 12^2.
%e a(719) = 42862647 since sigma(42862647) = 58003456 = 7616^2 and phi(42862627*719) = phi(30818243193) = 20210602896 = 142164^2.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t sigma[n_]:=DivisorSigma[1, n]
%t 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}]
%t 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 *)
%Y Cf. A000010, A000203, A000290, A259789, A259915.
%K nonn
%O 1,3
%A _Zhi-Wei Sun_, Jul 08 2015
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