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.

1, 85, 1, 273, 34, 85, 10, 364, 250, 17, 2, 2223, 204, 5, 34, 546, 10, 60, 680, 60, 10, 1, 5, 364, 48, 34, 40, 451, 136, 17, 10, 273, 2, 5, 2, 5089, 10570, 1020, 451, 10, 60, 5, 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

Offset: 1

Author

Zhi-Wei Sun, Jul 08 2015

Keywords

• nonn

Comments

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.

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.

Examples

			a(2) = 85 since phi(85) = 64 = 8^2 and sigma(85*2) = 324 = 18^2.
a(673) = 3451030792 since phi(3451030792) = 1564993600 = 39560^2 and sigma(3451030792*673) = sigma(2322543723016) = 4768807737600 = 2183760^2.
		

References

• 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.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..1000
• Zhi-Wei Sun, Checking the conjecture for r = a/b with a,b = 1..150
• Zhi-Wei Sun, Problems on combinatorial properties of primes, arXiv:1402.6641 [math.NT], 2014.

Crossrefs

Cf. A000010, A000203, A000290, A259789, A259916.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
sigma[n_]:=DivisorSigma[1,n]
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}]
(* Second program: *)
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 *)

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