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A259915 Least positive integer k such that phi(k) and sigma(k*n) are both squares, where phi(.) is Euler's totient function and sigma(m) is the sum of all positive divisors of m.

%I A259915 #25 May 05 2017 06:16:36
%S A259915 1,85,1,273,34,85,10,364,250,17,2,2223,204,5,34,546,10,60,680,60,10,1,
%T A259915 5,364,48,34,40,451,136,17,10,273,2,5,2,5089,10570,1020,451,10,60,5,
%U A259915 1970,114,114,17,2,4446,185,8,10,17,5,546,17,285,63,204,8,540,816,5,57,147744,2761,1,505,451,5,1
%N A259915 Least positive integer k such that phi(k) and sigma(k*n) are both squares, where phi(.) is Euler's totient function and sigma(m) is the sum of all positive divisors of m.
%C A259915 Conjecture: a(n) exists for any n > 0. In general, every positive rational number r can be written as m/n, where m and n are positive integers with phi(m) and sigma(n) both squares of integers.
%C A259915 For example, 4/5 = 136/170 with phi(136) = 8^2 and sigma(170) = 18^2, and 5/4 = 1365/1092 with phi(1365) = 24^2 and sigma(1092) = 56^2.
%D A259915 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 A259915 Zhi-Wei Sun, <a href="/A259915/b259915.txt">Table of n, a(n) for n = 1..1000</a>
%H A259915 Zhi-Wei Sun, <a href="/A259915/a259915_1.txt">Checking the conjecture for r = a/b with a,b = 1..150</a>
%H A259915 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 A259915 a(2) = 85 since phi(85) = 64 = 8^2 and sigma(85*2) = 324 = 18^2.
%e A259915 a(673) = 3451030792 since phi(3451030792) = 1564993600 = 39560^2 and sigma(3451030792*673) = sigma(2322543723016) = 4768807737600 = 2183760^2.
%t A259915 SQ[n_]:=IntegerQ[Sqrt[n]]
%t A259915 sigma[n_]:=DivisorSigma[1,n]
%t A259915 Do[k=0;Label[aa];k=k+1;If[SQ[EulerPhi[k]]&&SQ[sigma[k*n]],Goto[bb],Goto[aa]];Label[bb];Print[n, " ", k];Continue,{n,1,70}]
%t A259915 (* Second program: *)
%t A259915 Table[k = 1; While[Times @@ Boole@ Map[IntegerQ@ Sqrt@ # &, {EulerPhi@ k, DivisorSigma[1, k n]}] < 1, k++]; k, {n, 70}] (* _Michael De Vlieger_, May 04 2017 *)
%o A259915 (Perl) use ntheory ":all"; for my $n (1..100) { my $k = 1; $k++ until is_power(euler_phi($k),2) && is_power(divisor_sum($k*$n),2); say "$n $k" } # _Dana Jacobsen_, May 04 2017
%Y A259915 Cf. A000010, A000203, A000290, A259789, A259916.
%K A259915 nonn
%O A259915 1,2
%A A259915 _Zhi-Wei Sun_, Jul 08 2015