A114831 Each term is previous term plus floor of harmonic mean of two previous terms.

1, 2, 3, 5, 8, 14, 24, 41, 71, 122, 211, 365, 632, 1094, 1895, 3282, 5684, 9845, 17052, 29534, 51154, 88601, 153461, 265802, 460382, 797405, 1381145, 2392213, 4143434, 7176638, 12430301, 21529913, 37290903, 64589738, 111872708, 193769214, 335618123, 581307641, 1006854369, 1743922922

Offset

1,2

Comments

For two numbers x and y, HarmonicMean[x,y] = [(GeometricMean[x,y])^2] / Arithmetic Mean[x,y]. What is this sequence, asymptotically? a(n) is prime for n = 2, 3, 4, 6, 9, 15, ... are there an infinite number of prime values?

Links

Table of n, a(n) for n=1..40.

Eric Weisstein's World of Mathematics, Harmonic Mean.

Eric Weisstein's World of Mathematics, Geometric Mean.

Formula

a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + floor(HarmonicMean[a(n),a(n-1)]). a(n+1) = a(n) + floor[(2*a(n)*a(n-1))/(a(n)+a(n-1))].

Example

a(3) = 2 + floor(2*1*2/(1+2)) = 2 + floor(4/3) = 2 + 1 = 3.
a(4) = 3 + floor(2*2*3/(2+3)) = 3 + floor(12/5) = 3 + 2 = 5.
a(5) = 5 + floor(2*3*5/(3+5)) = 5 + floor(30/8) = 5 + 3 = 8.
a(6) = 8 + floor(2*5*8/(5+8)) = 8 + floor(80/13] = 8 + 6 = 14.
a(7) = 14 + floor(2*8*14/(8+14)) = 14 + floor(112/11) = 14 + 10 = 24.

Maple

hMean := proc(a, b)
    2*a*b/(a+b) ;
end proc:
A114831 := proc(n)
    option remember;
    if n<= 2 then
        n;
    else
        procname(n-1)+floor(hMean(procname(n-1), procname(n-2))) ;
    end if;
end proc:
seq(A114831(n), n=1..60) ; # R. J. Mathar, Jun 23 2014

Crossrefs

Cf. A065094, A065095.

Sequence in context: A086661 A018154 A340215 * A343161 A274110 A347018

Adjacent sequences: A114828 A114829 A114830 * A114832 A114833 A114834

Keyword

nonn,easy

Author

Jonathan Vos Post, Feb 19 2006

Extensions

Corrected by R. J. Mathar, Jun 23 2014

Typo in a(40) corrected by Seth A. Troisi, May 13 2022

Status

approved