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A245214
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Numbers k such that A245212(k) < 0.
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9
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144, 192, 216, 240, 288, 336, 360, 384, 432, 480, 504, 540, 576, 600, 648, 672, 720, 768, 792, 840, 864, 900, 936, 960, 1008, 1056, 1080, 1152, 1200, 1248, 1260, 1296, 1320, 1344, 1440, 1512, 1536, 1560, 1584, 1620, 1632, 1680, 1728, 1800, 1824, 1848, 1872, 1920, 1944, 1980, 2016, 2040, 2100, 2112, 2160, 2240
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OFFSET
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1,1
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COMMENTS
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If d are divisors of k then values of sequence A245212(k) are by bending moments in point 0 of static forces of sizes tau(d) operating in places d on the cantilever as the nonnegative number axis of length k with bracket in point 0 by the schema: A245212(k) = (k * tau(k)) - Sum_{(d<k) | k} (d * tau(d)).
Numbers k such that A038040(k) = k * tau(k) < A245211(k) = Sum_{(d<k) | k} (d * tau(d)).
Numbers whose divisors have a mean abundancy index that is larger than 2.
The numbers of terms that do not exceed 10^k, for k = 3, 4, ..., are 24, 243, 2571, 25583, 254794, 2551559, 25514104, 255112225, ... . Apparently, the asymptotic density of this sequence exists and equals 0.02551... .
The least odd term in this sequence is a(276918705) = 10854718875. (End)
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LINKS
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EXAMPLE
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Number 144 is in sequence because 144 * tau(144) = 2160 < Sum_{(d<144) | 144} (d * tau(d)) = 2226.
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MATHEMATICA
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f[p_, e_] := ((e+1)*p^2 - (e+2)*p + p^(-e))/((e+1)*(p-1)^2); s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Select[Range[2500], s[#] > 2 &] (* Amiram Eldar, Jul 19 2024 *)
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PROG
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(Magma) [n: n in [1..100000] | (2*(n*(#[d: d in Divisors(n)]))-(&+[d*#([e: e in Divisors(d)]): d in Divisors(n)])) lt 0]
(PARI) isok(n) = (n*numdiv(n) - sumdiv(n, d, (d<n)*d*numdiv(d))) < 0; \\ Michel Marcus, Aug 06 2014
(PARI) is(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i, 1]; e=f[i, 2]; (-2*p - e*p + p^2 + e*p^2 + p^(-e))/((e + 1)*(p - 1)^2)) > 2; } \\ Amiram Eldar, Jul 19 2024
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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