A006086 Unitary harmonic numbers (those for which the unitary harmonic mean is an integer). (Formerly M4248)

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

1,2

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.

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

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 *)

Prog

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

Crossrefs

See A006087 for more info.

Cf. A103339, A103340.

Sequence in context: A286325 A335387 A063947 * A273507 A131513 A204558

Adjacent sequences: A006083 A006084 A006085 * A006087 A006088 A006089

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from Emeric Deutsch, Dec 22 2004

Status

approved