OFFSET
1,1
COMMENTS
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)).
From Amiram Eldar, Jul 19 2024: (Start)
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)
LINKS
Jens Kruse Andersen, Table of n, a(n) for n = 1..10000
EXAMPLE
Number 144 is in sequence because 144 * tau(144) = 2160 < Sum_{(d<144) | 144} (d * tau(d)) = 2226.
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jaroslav Krizek, Jul 23 2014
STATUS
approved