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A114831
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Each term is previous term plus floor of harmonic mean of two previous terms.
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1
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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
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OFFSET
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1,2
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COMMENTS
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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?
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LINKS
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FORMULA
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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))].
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EXAMPLE
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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.
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MAPLE
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hMean := proc(a, b)
2*a*b/(a+b) ;
end proc:
option remember;
if n<= 2 then
n;
else
procname(n-1)+floor(hMean(procname(n-1), procname(n-2))) ;
end if;
end proc:
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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