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, 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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Jul 08 2015
STATUS
approved