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A319745
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Nonunitary harmonic numbers: numbers such that the harmonic mean of their nonunitary divisors is an integer.
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11
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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
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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MATHEMATICA
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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]
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PROG
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(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
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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