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A222264 Numbers n such that 2n/sigma(n) - 1 = 1/x for some integer x. 4
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 (list; graph; refs; listen; history; text; internal format)
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: A190896 A135772 A209064 * A051783 A033083 A302406

Adjacent sequences:  A222261 A222262 A222263 * A222265 A222266 A222267

KEYWORD

nonn

AUTHOR

M. F. Hasler, Feb 20 2013

STATUS

approved

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Last modified December 9 20:24 EST 2018. Contains 318023 sequences. (Running on oeis4.)