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

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

Author

N. J. A. Sloane

Keywords

• nonn
• nice

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)

Crossrefs

Cf. A006086, A034444, A034448, A077610.

Programs

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

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 *)

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

Formula

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

Extensions

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

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