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A006086
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Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).
(Formerly M4248)
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26
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1, 6, 45, 60, 90, 420, 630, 1512, 3780, 5460, 7560, 8190, 9100, 15925, 16632, 27300, 31500, 40950, 46494, 51408, 55125, 64260, 66528, 81900, 87360, 95550, 143640, 163800, 172900, 185976, 232470, 257040, 330750, 332640, 464940, 565488, 598500, 646425, 661500
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OFFSET
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1,2
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COMMENTS
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Let ud(n) and usigma(n) be number of and sum of unitary divisors of n; then the unitary harmonic mean of the unitary divisors is H(n) = n*ud(n)/usigma(n). - Emeric Deutsch, Dec 22 2004
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REFERENCES
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N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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If m is a term and omega(m) = A001221(m) = k, then m < 2^(k*2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020
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MATHEMATICA
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ud[n_] := 2^PrimeNu[n]; usigma[n_] := Sum[ If[ GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; uhm[n_] := n*ud[n]/usigma[n]; Reap[ Do[ If[ IntegerQ[uhm[n]], Print[n]; Sow[n]], {n, 1, 10^6}]][[2, 1]] (* Jean-François Alcover, May 16 2013 *)
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PROG
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(Haskell)
a006086 n = a006086_list !! (n-1)
a006086_list = filter ((== 1) . a103340) [1..]
(PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
isok(n) = my(v=udivs(n)); denominator(n*#v/vecsum(v))==1; \\ Michel Marcus, May 07 2017
(PARI) is(n, f=factor(n))=(n<<(#f~))%sumdivmult([n, f], d, if(gcd(d, n/d)==1, d))==0 \\ Charles R Greathouse IV, Nov 05 2021
(PARI) list(lim)=my(v=List()); forfactored(n=1, lim\1, if((n[1]<<omega(n))%sumdivmult(n, d, if(gcd(d, n[1]/d)==1, d))==0, listput(v, n[1]))); Vec(v) \\ Charles R Greathouse IV, Nov 05 2021
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CROSSREFS
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KEYWORD
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nonn,nice
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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