A335289 Nonsquarefree numbers that are both balanced numbers (A020492) and unitary balanced numbers (A335288).

1492260, 1741740, 2369640, 7192260, 83445180, 91798980, 104370420, 125214180, 141996120, 148532076, 162910980, 171175788, 196899780, 199793412, 201246660, 229849620, 297085860, 298993140, 398023080, 442859940, 540201480, 548305740, 796792920, 801375660, 835975140

Offset: 1

Author

Amiram Eldar, May 30 2020

Keywords

• nonn

Comments

The squarefree balanced numbers (A078557) are also unitary balanced numbers (A335288), since all the divisors of squarefree numbers are unitary, and thus if k is squarefree, then sigma(k) = usigma(k) and phi(k) = uphi(k).

Examples

			1492260 is a term since sigma(1492260)/phi(1492260) = 5806080/276480 = 21 is an integer, usigma(1492260)/uphi(1492260) = 4147200/414720 = 10 is an integer, and 1492260 is not squarefree since it is divisible by 4 = 2^2.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..4045 (calculated using data from _Jud McCranie_)

Crossrefs

Intersection of A013929, A020492 and A335288.

Cf. A000010 (phi), A000203 (sigma), A047994 (uphi), A034448 (usigma), A078557.

Programs

Mathematica

f1[1, 1] = 1; f1[p_, e_] := (p^(e+1) - 1)/p^(e-1)/(p-1)^2; f2[1, 1] = 1; f2[p_, e_] := (p^e + 1)/(p^e -1); balQ[n_] := And @@ IntegerQ /@ Times@@({f1[#1, #2], f2[#1, #2]}& @@@ FactorInteger[n]); Select[Range[3*10^6], !SquareFreeQ[#] && balQ[#] &]