A348923 Numbers that are both unitary and nonunitary harmonic numbers.

45, 60, 3780, 64260, 3112200, 6320160

Offset

1,1

Comments

a(7) > 10^12, if it exists.

For each term the two sets of unitary and nonunitary divisors both contain more than one element. The only number with a single unitary divisor is 1 which does not have nonunitary divisors. Numbers with a single nonunitary divisor are the squares of primes which are not unitary harmonic numbers. Therefore, this sequence is a subsequence of A348715.

Nonsquarefree numbers k such that A034448(k) divides k*A034444(k) and A048146(k) divides k*A048105(k). - Daniel Suteu, Nov 05 2021

Links

Table of n, a(n) for n=1..6.

Example

45 is a term since the unitary divisors of 45 are 1, 5, 9 and 45, and their harmonic mean is 3, and the nonunitary divisors of 45 are 3 and 15, and their harmonic mean is 5.

Mathematica

usigma[1] = 1; usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); Select[Range[65000], !SquareFreeQ[#] && IntegerQ[# * (d = 2^PrimeNu[#])/ (s = usigma[#])] && IntegerQ[# * (DivisorSigma[0, #] - d)/(DivisorSigma[1, #] - s)] &]

Crossrefs

Intersection of A006086 and A319745.

Subsequence of A348715.

Cf. A348922.

Cf. A034444, A034448, A048105, A048146.

Sequence in context: A060463 A206024 A151743 * A348922 A179007 A138046

Adjacent sequences: A348920 A348921 A348922 * A348924 A348925 A348926

Keyword

nonn,more

Author

Amiram Eldar, Nov 04 2021

Status

approved