A006087 Unitary harmonic means H(n) of the unitary harmonic numbers (A006086). (Formerly M0452)

1, 2, 3, 4, 4, 7, 7, 6, 9, 13, 10, 13, 10, 7, 11, 15, 10, 15, 9, 12, 7, 17, 12, 18, 16, 14, 19, 20, 19, 12, 15, 20, 10, 20, 18, 22, 19, 13, 12, 13, 17, 29, 18, 33, 20, 23, 29, 34, 23, 22, 31, 38, 24, 23, 38, 33, 37, 40, 19, 38, 24, 37, 29, 40, 22, 34, 24, 33

Offset

1,2

Comments

Let d(n) and sigma(n) be number and sum of unitary divisors of n; then unitary harmonic mean of unitary divisors is H(n)=n*d(n)/sigma(n).

Each term appears a finite number of times in the sequence (Hagis and Lord, 1975). - Amiram Eldar, Mar 10 2023

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

Peter Hagis, Jr. and Graham Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7.

Peter Hagis, Jr. and Graham Lord, Unitary harmonic numbers, Proc. Amer. Math. Soc., 51 (1975), 1-7. (Annotated scanned copy)

Formula

a(n) = A103339(A006086(n)). - Reinhard Zumkeller, Mar 17 2012

Maple

A034444 := proc(n) 2^nops(ifactors(n)[2]) ; end: A034448 := proc(n) local ans, i, ifs ; ans :=1 ; ifs := ifactors(n)[2] ; for i from 1 to nops(ifs) do ans := ans*(1+ifs[i][1]^ifs[i][2]) ; od ; RETURN(ans) ; end: A006086 := proc(n) n*A034444(n)/A034448(n) ; end: for n from 1 to 5000000 do uhn := A006086(n) : if type(uhn, 'integer') then printf("%d, ", uhn) ; fi ; od : # R. J. Mathar, Jun 06 2007

Mathematica

ud[n_] := 2^PrimeNu[n]; usigma[n_] := Sum[ If[ GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; a[n_] := n*ud[n]/usigma[n]; a[1] = 1; Reap[ Do[ If[ IntegerQ[h = a[n]], Print[h]; Sow[h]], {n, 1, 10^7}]][[2, 1]] (* Jean-François Alcover, May 16 2013 *)
uh[n_] := n * Times @@ (2/(1 + Power @@@ FactorInteger[n])); uh[1] = 1; Select[Array[uh, 10^6], IntegerQ] (* Amiram Eldar, Mar 10 2023 *)

Prog

(PARI) {ud(n)=2^omega(n)} {sud(n) = sumdiv(n, d, if(gcd(d, n/d)==1, d))} {H(n)=n*ud(n)/sud(n)} for(n=1, 10000000, if(((n*ud(n))%sud(n))==0, print1(H(n)", "))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008
(PARI) uhmean(n) = {my(f = factor(n)); n*prod(i=1, #f~, 2/(1+f[i, 1]^f[i, 2])); };
lista(kmax) = {my(uh); for(k = 1, kmax, uh = uhmean(k); if(denominator(uh) == 1, print1(uh, ", "))); } \\ Amiram Eldar, Mar 10 2023
(Haskell)
import Data.Ratio ((%), numerator, denominator)
a006087 n = a006087_list !! (n-1)
a006087_list = map numerator $ filter ((== 1) . denominator) $
   map uhm [1..]  where uhm n = (n * a034444 n) % (a034448 n)
-- Reinhard Zumkeller, Mar 17 2012

Crossrefs

Cf. A006086, A034444, A034448, A077610.

Sequence in context: A244903 A342337 A167932 * A364675 A241315 A136330

Adjacent sequences: A006084 A006085 A006086 * A006088 A006089 A006090

Keyword

nonn,nice

Author

N. J. A. Sloane

Extensions

More terms from R. J. Mathar, Jun 06 2007

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Mar 02 2008

Status

approved