[seqfan] Re: A conjeture about 34155

Psychedelic Geometry psychgeometry at gmail.com
Sun Apr 15 00:38:28 CEST 2012

Hello...

sigma(n) = Sum(d|n, d) = g(n) + f(n) + n

where:
g(n) = Sum(d|n, d<=sqrt(n)) >= 1 (for n>1)
f(n) = Sum(d|n, d>sqrt(n) and d!=n) >= 1 (for n>1)

if n is in A the set of integers that are equal to the sum of their proper
divisors, greater than their square root, then f(n)=n

This means that: sigma(n) = g(n) + 2n and the abundance function is  s(n) =
g(n) >= 1

and then the elements in A can not be deficient or perfect.

This simple thing limitates the search of odd elements to the sequence:
A005231

And using my super-mega-mathematica package, that nobody uses, and that can
be found here:

I have coded the following...

<< OEIS`
OEISFunction["A005231"]
f[n_]:=Plus@@Select[Drop[Divisors[n], -1], Sqrt[n]<#&]
Select[Table[A005231[n], {n, 1, 1000}], f[#] == # &]

This gives {34155}, and because A005231(1000) = 492975, there are no other
odd elements below this mark.

The abundances of the elements of A <10^6 are {12, 56, 810, 992} , (the
value 810 only happens for 34155)

DeleteDuplicates[(DivisorSigma[1, #] - 2 #) & /@
Select[Table[Range[n], {n, 1, 10^3}], f[#] == # &]]

CONJECTURE-1: The abundance in A is >=12.

If we take a close look at https://oeis.org/A141545 we can see that all
elements in A141545 have omega(n)=3 except {24, 54, 304, 127744, ...}
that have omega(n)=2.

CONJECTURE-2: All elements in A141545 have 1< omega(n) <=3.

CONJECTURE-3: All integers with abundance(n) = 12 and omega(n)=3, are in A

More tomorrow...

Saludos de Enrique.

-----Mensaje original-----
De: seqfan-bounces at list.seqfan.eu
[mailto:seqfan-bounces at list.seqfan.eu]En nombre de Claudio Meller
Para: Sequence Fanatics Discussion list
Asunto: [seqfan] A conjeture about 34155

Hi seqfans, one of my blog readers, Jordi Domènech i Arnau, has sent me
this :

*I have looked for the numbers that are equal to the sum of their proper
> divisors, greater than their square root. *

*I found a lot of even numbers with this property, but  I´ve only found
>  one odd number, 34155, with this property.*

He makes the conjeture that 34155 is the only odd number with this
caracteristic.

34155 = 207+253+297+345+495+621+759+1035+1265+1485+2277+3105+3795+6831+11385

I ´ve made the sequence (not in the OEIS):
Numbers N  equal to the sum of its proper divisors greater than sqr (N)

42, 54, 66, 78, 102, 114, 138, 174, 186, 222, 246, 258, 282, 318, 354, 366,
402, 426, 438, 474, 498, 534, 582, 606, 618, 642, 654, 678, 762, 786, 812,
822, 834, 868, 894, 906, 942, 978, 1002, 1036, 1038, 1074, 1086, 1146,
1148, 1158, 1182, 1194, 1204, 1266, 1316, 1338, 1362, 1372, 1374, 1398,
1434, 1446, 1484, 1506, 1542, 1578, 1614, 1626, 1652, 1662, 1686, 1698,
1708, 1758, 1842, 1866, 1876, 1878, 1902, 1986, 1988, 2022, 2044, 2082,
2094, 2118, 2154, 2202, 2212, 2238, 2274, 2298, 2324, 2334, 2382, 2406,
2454, 2492, 2514, 2526, 2586, 2598, 2634, 2658, 2694, 2716, 2742, 2766,
2778, 2802, 2828, 2874, 2884, 2922, 2946, 2994, 2996, 3018, 3052, 3054,
3126, 3138, 3164, 3246, 3282, 3342, 3378, 3414, 3426, 3462

This sequence has to many terms of  A141545 (Numbers n whose abundance is
12)

Can anybody prove this or find a counterexample?
Do you find this interesting?

Thanks
--
Claudio Meller
http://grageasdefarmacia.blogspot.com
http://todoanagramas.blogspot.com/
http://simplementenumeros.blogspot.com/

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