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

1,2

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.

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.

Example

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.

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 *)

Prog

(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

Crossrefs

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

Sequence in context: A111194 A096132 A078083 * A317438 A105328 A033405

Adjacent sequences: A259912 A259913 A259914 * A259916 A259917 A259918

Keyword

nonn

Author

Zhi-Wei Sun, Jul 08 2015

Status

approved