A319745 - OEIS

%I #14 Jan 05 2025 19:51:41

%S 4,9,12,18,24,25,45,49,54,60,112,121,126,150,168,169,270,289,294,336,

%T 361,529,560,594,637,726,841,961,1014,1232,1369,1638,1680,1681,1734,

%U 1849,1984,2166,2184,2209,2430,2520,2688,2700,2809,2850,3174,3481,3721,3780

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

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

%C Supersequence of A064591 (nonunitary perfect numbers).

%C 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).

%H Amiram Eldar, <a href="/A319745/b319745.txt">Table of n, a(n) for n = 1..10000</a>

%H Steve Ligh and Charles R. Wall, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/25-4/ligh.pdf">Functions of Nonunitary Divisors</a>, Fibonacci Quarterly, Vol. 25 (1987), pp. 333-338.

%t 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]

%o (PARI) hm(v) = #v/sum(k=1, #v, 1/v[k]);

%o vnud(n) = select(x->(gcd(x, n/x)!=1), divisors(n));

%o isok(n) = iferr(denominator(hm(vnud(n))) == 1, E, 0); \\ _Michel Marcus_, Oct 28 2018

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

%K nonn

%O 1,1

%A _Amiram Eldar_, Sep 27 2018