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A006086 Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).

%I A006086 M4248 #53 Jan 05 2025 19:51:33
%S A006086 1,6,45,60,90,420,630,1512,3780,5460,7560,8190,9100,15925,16632,27300,
%T A006086 31500,40950,46494,51408,55125,64260,66528,81900,87360,95550,143640,
%U A006086 163800,172900,185976,232470,257040,330750,332640,464940,565488,598500,646425,661500
%N A006086 Unitary harmonic numbers (those for which the unitary harmonic mean is an integer).
%C A006086 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
%C A006086 A103340(a(n)) = 1; A103339(a(n)) = A006087(n). - _Reinhard Zumkeller_, Mar 17 2012
%D A006086 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006086 Donovan Johnson, <a href="/A006086/b006086.txt">Table of n, a(n) for n = 1..290</a> (terms < 10^12)
%H A006086 Takeshi Goto, <a href="http://doi.org/10.1216/rmjm/1194275935">Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers</a>, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
%H A006086 P. Hagis, Jr. and G. Lord, <a href="https://doi.org/10.1090/S0002-9939-1975-0369231-9">Unitary harmonic numbers</a>, Proc. Amer. Math. Soc., 51 (1975), 1-7.
%H A006086 P. Hagis, Jr. and G. Lord, <a href="/A006086/a006086.pdf">Unitary harmonic numbers</a>, Proc. Amer. Math. Soc., 51 (1975), 1-7. (Annotated scanned copy)
%H A006086 Charles R. Wall, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/21-1/wall.pdf">Unitary harmonic numbers</a>, Fibonacci Quarterly, Vol. 21, No. 1 (1983), pp. 18-25.
%F A006086 If m is a term and omega(m) = A001221(m) = k, then m < 2^(k*2^k) (Goto, 2007). - _Amiram Eldar_, Jun 06 2020
%t A006086 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 *)
%o A006086 (Haskell)
%o A006086 a006086 n = a006086_list !! (n-1)
%o A006086 a006086_list = filter ((== 1) . a103340) [1..]
%o A006086 -- _Reinhard Zumkeller_, Mar 17 2012
%o A006086 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A006086 isok(n) = my(v=udivs(n)); denominator(n*#v/vecsum(v))==1; \\ _Michel Marcus_, May 07 2017
%o A006086 (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
%o A006086 (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
%Y A006086 See A006087 for more info.
%Y A006086 Cf. A103339, A103340.
%K A006086 nonn,nice
%O A006086 1,2
%A A006086 _N. J. A. Sloane_
%E A006086 More terms from _Emeric Deutsch_, Dec 22 2004