A222264 - OEIS

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

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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