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

1,1

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).

Links

Amiram Eldar, Table of n, a(n) for n = 1..4045 (calculated using data from Jud McCranie)

Example

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.

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[#] &]

Crossrefs

Intersection of A013929, A020492 and A335288.

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

Sequence in context: A126175 A128837 A297881 * A228761 A206060 A251973

Adjacent sequences: A335286 A335287 A335288 * A335290 A335291 A335292

Keyword

nonn

Author

Amiram Eldar, May 30 2020

Status

approved