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%I #22 Jan 16 2023 15:41:09
%S 1,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,
%T 97,101,103,107,109,113,120,127,131,137,139,149,151,157,163,167,173,
%U 179,180,191,193,197,199,211,223,227,229,233,239,240,269,271,277,281,283,293,307
%N Highly imperfect numbers: n sets a record for the value of abs(sigma(n)-2*n) (absolute value of A033879).
%C A perfect number n is defined by sigma(n) = 2n, so the value of i(n) = |sigma(n)-2n| measures the degree of perfection of n. The larger i(n) is, the more "imperfect" n is. I call the numbers n such that i(k) < i(n) for all k < n "highly-imperfect numbers".
%C RECORDS transform of |A033879|.
%C Initial terms are odd primes but then even numbers appear.
%C The last odd term is a(79) = 719. (Proof: sigma(27720n) >= 11080n, and so sigma(27720n) >= 4 * 27720(n + 1) for n >= 8, so there is no odd member of this sequence between 27720 * 8 and 27720 * 9, between 27720 * 9 and 2770 * 10, etc.; the remaining terms are checked by computer.) [_Charles R Greathouse IV_, Apr 12 2010]
%H R. J. Mathar and Donovan Johnson, <a href="/A074918/b074918.txt">Table of n, a(n) for n = 1..1000</a> (first 238 terms from R. J. Mathar)
%H Joseph L. Pe, <a href="http://www.primepuzzles.net/puzzles/puzz_241.htm">Puzzle 241. Highly imperfect primes</a>, and answers, on C. Rivera's primepuzzles.net. [From _M. F. Hasler_, May 31 2009]
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%t r = 0; l = {}; Do[ n = Abs[2 i - DivisorSigma[1, i]]; If[n > r, r = n; l = Append[l, i]], {i, 1, 10^4}]; l
%t DeleteDuplicates[Table[{n,Abs[DivisorSigma[1,n]-2n]},{n,350}],GreaterEqual[ #1[[2]],#2[[2]]]&][[All,1]] (* _Harvey P. Dale_, Jan 16 2023 *)
%Y Cf. A033879, A075728.
%K easy,nice,nonn
%O 1,2
%A _Joseph L. Pe_, Oct 01 2002
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