A293391 Integers n such that sigma(n)/phi(n) is a perfect square.

1, 14, 30, 105, 248, 264, 418, 714, 1485, 3080, 3135, 3596, 3828, 3956, 4064, 5396, 6678, 8636, 10098, 12648, 20026, 20790, 21318, 22152, 23374, 24882, 25714, 26040, 35074, 35343, 39105, 41656, 43890, 44660, 49938, 55154, 56134, 56536, 61344, 71145, 74613, 86304, 87087, 94944

Offset: 1

Author

Jose Arnaldo Bebita Dris, Oct 08 2017

Keywords

• nonn

Comments

From Robert Israel, Dec 12 2017: (Start)

Intersection of A011257 and A020492.

If x and y are coprime members of the sequence, then x*y is in the sequence.

Contains all members of A133028 except 3. (End)

Examples

			sigma(14)=3*8=24, phi(14)=14*(1/2)*(6/7)=6, sigma(14)/phi(14)=2^2, so 14 is in the list.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 349 terms from Robert Israel)
• J. A. B. Dris and C. Leibovici, Why is this sequence not in the OEIS?, October 8 2017.
• ProofWiki, Integers whose ratio between sigma and phi is square, misses the second term, 14 as of Dec 2017.

Crossrefs

Cf. A000010, A000203, A011257, A020492, A133028, A289336, A289412, A292422.

Programs

Maple

for n from 1 to 100000 do
    r := numtheory[sigma](n)/numtheory[phi](n) ;
    if issqr(r) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 07 2017

Mathematica

Select[Range[10^5], IntegerQ@ Sqrt[DivisorSigma[1, #]/EulerPhi[#]] &] (* Michael De Vlieger, Dec 08 2017 *)

PARI

isok(n) = my(q=sigma(n)/eulerphi(n)); issquare(q) && (denominator(q) == 1); \\ Michel Marcus, Dec 07 2017; corrected Sep 21 2019

Formula

a(n) = sigma(n)/phi(n) = m^2, for some integer m.