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Search: id:A114832
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| A114832 |
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Each term is previous term plus ceiling of harmonic mean of two previous terms. |
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1
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| 1, 2, 4, 7, 13, 23, 40, 70, 121, 210, 364, 631, 1093, 1894, 3281, 5683, 9844, 17050, 29532, 51151, 88597, 153455, 265792, 460366, 797377, 1381098, 2392132, 4143295, 7176398, 12429886, 21529195, 37289660, 64587586, 111868981, 193762759
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OFFSET
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1,2
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COMMENT
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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, 4, 5, 6, 12, ... are there an infinite number of prime values?
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LINKS
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Eric Weisstein's World of Mathematics, Harmonic Mean.
Eric Weisstein's World of Mathematics, Geometric Mean.
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FORMULA
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a(1) = 1, a(2) = 2, for n>2: a(n+1) = a(n) + ceiling(HarmonicMean[a(n),a(n-1)]). a(n+1) = a(n) + ceiling[(2*a(n)*a(n-1))/(a(n)+a(n-1))].
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EXAMPLE
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a(3) = 2 + ceiling[2*1*2/(1+2)] = 2 + ceiling[4/3] = 2 + 2 = 4.
a(4) = 4 + ceiling[2*2*4/(2+4)] = 4 + ceiling[16/6] = 4 + 3 = 7.
a(5) = 7 + ceiling[2*4*7/(4+7)] = 7 + ceiling[56/8] = 7 + 6 = 13.
a(6) = 13 + ceiling[2*7*13/(7+13)] = 13 + ceiling[182/13] = 13 + 10 = 23.
a(7) = 23 + ceiling[2*13*23/(13+23)] = 23 + ceiling[598/36] = 23 + 17 = 40.
a(8) = 40 + ceiling[2*23*40/(23+40)] = 40 + ceiling[1840/63] = 40 + 30 = 70.
a(9) = 70 + ceiling[2*40*70/(40+70)] = 70 + ceiling[5600/110] = 70 + 51 = 121.
a(10) = 121 + ceiling[2*70*121/(70+121)] = 121 + ceiling[16940/191] = 121 + 89 = 210.
a(11) = 210 + ceiling[2*121*210/(121+210)] = 121 + ceiling[50820/331] = 210 + 154 = 364.
a(12) = 364 + ceiling[2*210*364/(210+364)] = 364 + ceiling[152880/574] = 364 + 267 = 631.
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MAPLE
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a[1]:=1: a[2]:=2: for n from 2 to 40 do a[n+1]:=a[n]+ceil((2*a[n]*a[n-1])/(a[n]+a[n-1])) od: seq(a[n], n=1..40); - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 03 2006
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CROSSREFS
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Cf. A065094, A065095.
Adjacent sequences: A114829 A114830 A114831 this_sequence A114833 A114834 A114835
Sequence in context: A073832 A051013 A128609 this_sequence A136299 A003116 A078038
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KEYWORD
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easy,nonn
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AUTHOR
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Jonathan Vos Post (jvospost2(AT)yahoo.com), Feb 19 2006
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EXTENSIONS
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More terms from Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 03 2006
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