A222264 Numbers n such that 2n/sigma(n) - 1 = 1/x for some integer x.

1, 2, 3, 4, 8, 10, 14, 15, 16, 32, 44, 64, 110, 128, 135, 136, 152, 182, 184, 190, 248, 256, 315, 512, 585, 752, 819, 884, 1012, 1024, 1155, 1365, 1485, 1550, 1892, 2048, 2144, 2272, 2295, 2528, 4064, 4096, 4455, 6490, 7030, 8192, 8384, 8648, 9009, 9405, 9945

Offset

1,2

Comments

If the number x is a prime which does not divide n, then n*x is a perfect number. This happens (so far) only when x = 2n-1 = 2^p-1 is a Mersenne prime (cf. A000043). But if x does not divide n, as, e.g., for (n,x)=(10,9), then n*x is a so-called freestyle perfect number, cf. A058007: Namely it "would be perfect if x is assumed to be prime", which means that sigma(n*x) is replaced by sigma(n)*(x+1) in the relation 2P=sigma(P) characterizing perfect numbers P, listed in A000396.

See also the (more interesting) subsequence of odd terms, A222263.

Links

Donovan Johnson, Table of n, a(n) for n = 1..1000

Example

8 is in the sequence because 2 * 8/sigma(8) - 1 = 16/15 - 1 = 1/15.
9 is not in the sequence because 2 * 9/sigma(9) - 1 = 5/13.
10 is in the sequence because 2 * 10/sigma(10) - 1 = 20/18 - 1 = 1/9.

Mathematica

Select[Range[10^5], IntegerQ[2#/DivisorSigma[1, #] - 1] &] (* Alonso del Arte, Feb 20 2013 *)

Prog

(PARI) for(n=1, 9e9, numerator(2*n/sigma(n)-1)==1 & print1(n", "))

Crossrefs

Sequence in context: A365018 A135772 A209064 * A051783 A033083 A302406

Adjacent sequences: A222261 A222262 A222263 * A222265 A222266 A222267

Keyword

nonn

Author

M. F. Hasler, Feb 20 2013

Status

approved