A074918 Highly imperfect numbers: n sets a record for the value of abs(sigma(n)-2*n) (absolute value of A033879).

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

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 "highly-imperfect numbers".

RECORDS transform of |A033879|.

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

R. J. Mathar and Donovan Johnson, Table of n, a(n) for n = 1..1000 (first 238 terms from R. J. Mathar)

Joseph L. Pe, Puzzle 241. Highly imperfect primes, and answers, on C. Rivera's primepuzzles.net. [From M. F. Hasler, May 31 2009]

N. J. A. Sloane, Transforms

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

Cf. A033879, A075728.

Sequence in context: A093893 A056912 A075763 * A175524 A073579 A006005

Adjacent sequences: A074915 A074916 A074917 * A074919 A074920 A074921

Keyword

easy,nice,nonn

Author

Joseph L. Pe, Oct 01 2002

Status

approved