A078557 -id:A078557 - cp's OEIS frontend

A121556 Numbers such that sigma(n)^2 is divisible by UnitarySigma(n)*UnitaryPhi(n).

1, 2, 3, 6, 14, 15, 30, 35, 42, 70, 78, 105, 190, 210, 348, 357, 418, 570, 714, 910, 1045, 1144, 1254, 2090, 2296, 2730, 3135, 3432, 4060, 4522, 4674, 5278, 6270, 6888, 10168, 10659, 10824, 12180, 12441, 13566, 14630, 15834, 16770, 17160

Offset: 1

Yasutoshi Kohmoto, Sep 12 2006

nonn

Donovan Johnson, Table of n, a(n) for n = 1..1000

Cf. A000203, A121647, A078557, A020492, A034448, A047994.

for n from 1 to 20000 do if numtheory[sigma](n)^2 mod (A047994(n)*A034448(n)) = 0 then printf("%d,",n) ; end if;end do:

f[p_, e_] := (p^(e+1)-1)^2/(p-1)^2/(p^(2*e)-1); seqQ[1] = True; seqQ[n_] := IntegerQ [Times @@ (f @@@ FactorInteger[n])]; Select[Range[17160], seqQ] (* Amiram Eldar, Dec 11 2019 *)

A335288 Unitary balanced numbers: numbers k such that uphi(k) (A047994) divides usigma(k) (A034448).

Original entry on oeis.org

1, 2, 3, 6, 14, 15, 30, 35, 42, 44, 60, 70, 78, 105, 126, 132, 190, 210, 220, 312, 357, 418, 558, 570, 660, 693, 714, 728, 910, 1045, 1254, 1386, 1395, 1428, 1540, 2090, 2108, 2184, 2730, 2790, 3135, 3465, 3640, 3692, 3762, 4522, 4620, 4674, 5236, 5278, 6270

Offset: 1

Amiram Eldar, May 30 2020

nonn

Terms that are also balanced numbers (A020492) include the squarefree balanced numbers (A078557). The nonsquarefree common terms are in A335289.

			6 is a term since usigma(6) = 12 is divisible by uphi(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The unitary version of A020492.
A078557 is a subsequence.
Cf. A034448, A047994, A335289.

f[1, 1] = 1; f[p_, e_]:= (p^e + 1)/(p^e -1); ubalQ[n_] := IntegerQ[Times @@ (f @@@ FactorInteger[n])]; Select[Range[10^4], ubalQ]

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

Original entry on oeis.org

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

Amiram Eldar, May 30 2020

nonn

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).

			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.

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

Intersection of A013929, A020492 and A335288.

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

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[#] &]

A386573 Sum of the squarefree balanced divisors of n.

Original entry on oeis.org

1, 3, 4, 3, 1, 12, 1, 3, 4, 3, 1, 12, 1, 17, 19, 3, 1, 12, 1, 3, 4, 3, 1, 12, 1, 3, 4, 17, 1, 57, 1, 3, 4, 3, 36, 12, 1, 3, 4, 3, 1, 68, 1, 3, 19, 3, 1, 12, 1, 3, 4, 3, 1, 12, 1, 17, 4, 3, 1, 57, 1, 3, 4, 3, 1, 12, 1, 3, 4, 122, 1, 12, 1, 3, 19, 3, 1, 90, 1, 3, 4, 3, 1, 68, 1, 3, 4, 3, 1, 57, 1, 3, 4, 3, 1, 12, 1, 17, 4, 3

Offset: 1

Wesley Ivan Hurt, Jul 26 2025

nonn

Inverse Möbius transform of n * c(n) * mu(n)^2, where c = A351114.

Cf. A005117 (squarefree numbers), A020492 (balanced numbers), A078557, A351114, A386574.

Table[Sum[d (1 - Ceiling[DivisorSigma[1, d]/EulerPhi[d]] + Floor[DivisorSigma[1, d]/EulerPhi[d]]) MoebiusMu[d]^2, {d, Divisors[n]}], {n, 100}]

a(n) = Sum_{d|n} d * c(d) * mu(d)^2, where c = A351114.

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A121556 Numbers such that sigma(n)^2 is divisible by UnitarySigma(n)*UnitaryPhi(n).

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A335288 Unitary balanced numbers: numbers k such that uphi(k) (A047994) divides usigma(k) (A034448).

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

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A386573 Sum of the squarefree balanced divisors of n.

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