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

Author

Joseph L. Pe, Oct 01 2002

Keywords

• easy
• nice
• nonn

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

Crossrefs

Cf. A033879, A075728.

Programs

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 *)