The On-Line Encyclopedia of Integer Sequences (OEIS)

A006087 revision #35

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

(list; graph; refs; listen; history; text; internal format)

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).

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

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)

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 *)

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

(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