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

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• nice

Comments

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

A103340(a(n)) = 1; A103339(a(n)) = A006087(n). - Reinhard Zumkeller, Mar 17 2012

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Donovan Johnson, Table of n, a(n) for n = 1..290 (terms < 10^12)
• Takeshi Goto, Upper Bounds for Unitary Perfect Numbers and Unitary Harmonic Numbers, Rocky Mountain Journal of Mathematics, Vol. 37, No. 5 (2007), pp. 1557-1576.
• P. Hagis, Jr. and G. Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7.
• P. Hagis, Jr. and G. Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7. (Annotated scanned copy)
• Charles R. Wall, Unitary harmonic numbers, Fibonacci Quarterly, Vol. 21, No. 1 (1983), pp. 18-25.

Crossrefs

See A006087 for more info.

Cf. A103339, A103340.

Programs

Haskell

a006086 n = a006086_list !! (n-1)
a006086_list = filter ((== 1) . a103340) [1..]
-- Reinhard Zumkeller, Mar 17 2012

Mathematica

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 *)

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]<Charles R Greathouse IV, Nov 05 2021

Formula

If m is a term and omega(m) = A001221(m) = k, then m < 2^(k*2^k) (Goto, 2007). - Amiram Eldar, Jun 06 2020

Extensions

More terms from Emeric Deutsch, Dec 22 2004