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

Author

Amiram Eldar, Sep 27 2018

Keywords

• nonn

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.

Crossrefs

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

Programs

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]

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