A319745 Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.

4, 9, 12, 18, 24, 25, 45, 49, 54, 60, 112, 121, 126, 150, 168, 169, 270, 289, 294, 336, 361, 529, 560, 594, 637, 726, 841, 961, 1014, 1232, 1369, 1638, 1680, 1681, 1734, 1849, 1984, 2166, 2184, 2209, 2430, 2520, 2688, 2700, 2809, 2850, 3174, 3481, 3721, 3780

Offset

1,1

Comments

Includes all the numbers with a single nonunitary divisor. Those with more than one: 12, 18, 24, 45, 54, 60, 112, ...

Supersequence of A064591 (nonunitary perfect numbers).

Ligh & Wall showed that if p, 2p-1 and 2^p-1 are distinct primes (A172461, except for 2), then the following numbers are in the sequence: 6*p^2, p^2*(2p-1), 6*p^2*(2p-1), 2^(p+1)*3*(2^p-1), 2^(p+1)*15*(2^p-1) and 2^(p+1)*(2p-1)*(2^p-1).

Links

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

Steve Ligh and Charles R. Wall, Functions of Nonunitary Divisors, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

Mathematica

nudiv[n_] := Block[{d = Divisors[n]}, Select[d, GCD[#, n/#] > 1 &]]; nhQ[n_]:= Module[ {divs=nudiv[n]}, Length[divs] > 0 && IntegerQ[HarmonicMean[divs]]]; Select[Range[30000], nhQ]

Prog

(PARI) hm(v) = #v/sum(k=1, #v, 1/v[k]);
vnud(n) = select(x->(gcd(x, n/x)!=1), divisors(n));
isok(n) = iferr(denominator(hm(vnud(n))) == 1, E, 0); \\ Michel Marcus, Oct 28 2018

Crossrefs

Cf. A001599, A006086, A063947, A064591, A172461, A286325.

Sequence in context: A312864 A326071 A357497 * A312865 A376340 A348918

Adjacent sequences: A319742 A319743 A319744 * A319746 A319747 A319748

Keyword

nonn

Author

Amiram Eldar, Sep 27 2018

Status

approved