How rare is 13?

jm bergot

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Feb 8, 2011, 2:01:59 PM2/8/11

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To stimulate the economy I will use '$(n)' to stand for
sigma(n) for the sum of the divisors of n.
For three consecutive numbers abc, b| $(a) + $(c).
One sees this for 12, 13, 14 with $(12) + $(14)=28+24=
52= 4*13. One could feed this into OEIS (if it is not
already there) to help it reach 200,000 sequences.

Robert Israel

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Feb 8, 2011, 7:23:56 PM2/8/11

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The members of the sequence up to 10000 are
1, 13, 77, 489, 557, 1101, 1431, 2409, 8897.
It seems not to be in the OEIS.
Heuristically, the probability that sigma(x-1) + sigma(x+1) == 0 mod x
should be roughly 1/x, so we'd expect roughly log(n) members up to n.
--
Robert Israel isr...@math.MyUniversitysInitials.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

jm bergot

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Feb 9, 2011, 1:54:33 PM2/9/11

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The splendid numbers from R. Israel are gratefully
accepted in theee Paradise, Victoria, BC. One
sees some clumpiness in them--I wonder if this
continues. If you have been away from OEIS,
you can use this item as a triumphant return.

Valeri Astanoff

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Feb 10, 2011, 5:36:46 AM2/10/11

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Good day,

In case you intend to publish it in the OEIS,
here is a mathematica formula to generate
the first terms of your sequence:

f[n_]:=Mod[Tr@Divisors[n-1]+Tr@Divisors[n+1],n]==0;
Select[Range[1000000], f]

{13, 77, 489, 557, 1101, 1431, 2409, 8897, 538209}

--
Valeri

Valeri Astanoff

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Feb 10, 2011, 6:28:39 AM2/10/11

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Two more :

{13, 77, 489, 557, 1101, 1431, 2409, 8897, 538209, 2263024, 8910721}

V.

Valeri Astanoff

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Feb 10, 2011, 10:50:47 AM2/10/11

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> V.- Masquer le texte des messages précédents -
>
> - Afficher le texte des messages précédents -

Up to 100,000,000 :

{13, 77, 489, 557, 1101, 1431, 2409, 8897, 538209, 2263024, 8910721,
13685781, 17428321}

( none between 17,428,321 and 100,000,000 )

V.

jm bergot

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Feb 10, 2011, 1:40:51 PM2/10/11

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Thanks for the splendid production of solutions
from V. Astanoff. It is YOU who deserve the
credit for the work and should be the one to
toss this into the hopper at OEIS to aid them
in reaching their goal of 200,000 sequences.
Hurry while there is still room for this to fit!

Valeri Astanoff

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Feb 10, 2011, 3:45:48 PM2/10/11

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Bonsoir,

The most difficult is not to get an OEIS sequence number,
it's to convince Mr Sloane that it's interesting
enough to be approved and published !
Just one more word before I leave this subject :
I dare disagree with Prof. Israel about 1 being the first
number of the sequence, because, 0 having no divisors
(or an infinity), its sum of divisors is undefined.

Ecrit à Paris le 10 février à 21h45.
Au revoir.

--
Valeri

Richard Tobin

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Feb 10, 2011, 8:10:30 PM2/10/11

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In article <fdc41909-b068-43bf...@c10g2000vbv.googlegroups.com>,
Valeri Astanoff <asta...@gmail.com> wrote:

>Up to 100,000,000 :
>
>{13, 77, 489, 557, 1101, 1431, 2409, 8897, 538209, 2263024, 8910721,
>13685781, 17428321}

I confirm those and find no others below 500,000,000.

-- Richard

Richard Tobin

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Feb 10, 2011, 9:09:14 PM2/10/11

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In article <ij22a6$2917$1...@automatic.inf.ed.ac.uk>, I wrote:

>>{13, 77, 489, 557, 1101, 1431, 2409, 8897, 538209, 2263024, 8910721,
>>13685781, 17428321}

>I confirm those and find no others below 500,000,000.

The next is 962402769 and there are no others below 1,000,000,000.

-- Richard

Richard Tobin

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Feb 11, 2011, 6:50:44 AM2/11/11

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In article <ij25oa$2ac4$1...@automatic.inf.ed.ac.uk>, I wrote:

>>>{13, 77, 489, 557, 1101, 1431, 2409, 8897, 538209, 2263024, 8910721,
>>>13685781, 17428321}

>The next is 962402769 and there are no others below 1,000,000,000.

And I find none between 10^9 and 2x10^9, at which point I am giving up.

-- Richard