

A074918


Highly imperfect numbers: n sets a record for the value of abs(sigma(n)2*n) (absolute value of A033879).


5



1, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 120, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 180, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 240, 269, 271, 277, 281, 283, 293, 307
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OFFSET

1,2


COMMENTS

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 "highlyimperfect numbers".
Initial terms are odd primes but then even numbers appear.
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]


LINKS



MATHEMATICA

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
DeleteDuplicates[Table[{n, Abs[DivisorSigma[1, n]2n]}, {n, 350}], GreaterEqual[ #1[[2]], #2[[2]]]&][[All, 1]] (* Harvey P. Dale, Jan 16 2023 *)


CROSSREFS



KEYWORD

easy,nice,nonn


AUTHOR



STATUS

approved



