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The floor(1.001*x)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.
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%I #14 Sep 26 2019 09:05:30

%S 6,28,496,32445,130304,388076,199272950

%N The floor(1.001*x)-perfect numbers, where f-perfect numbers for an arithmetical function f are defined in A066218.

%C The floor(n)-perfect numbers are the ordinary perfect numbers. The first three floor[1.001x]-perfect numbers are also ordinary perfect numbers and the first discrepancy comes at the fourth term, 32445 (the fourth perfect number is 8128). Consider other coefficients > 1 but < 1.001. There is some kind of continuity working here. The first discrepancies, if they exist, come at later and later terms as these coefficients are made closer to 1.

%H J. Pe, <a href="http://jlpe.tripod.com/gpn/fperfect.htm">On a Generalization of Perfect Numbers</a>, J. Rec. Math., 31(3) (2002-2003), 168-172.

%e Let f(n) = floor(1.001*n). Then f(6) = 6 = 3+2+1 = f(3)+f(2)+f(1); so 6 is a term of the sequence.

%t f[x_] := Floor[1.001*x]; Select[ Range[1, 10^5], 2 * f[ # ] == Apply[ Plus, Map[ f, Divisors[ # ] ] ] & ]

%Y Cf. A066218.

%K nonn,more

%O 1,1

%A _Joseph L. Pe_, Dec 19 2001

%E a(5)-a(7) from _Amiram Eldar_, Sep 26 2019