[seqfan] Re: A conjeture about 34155

[Claudio Meller]

Hi seqfans, Manuel Valdivia send me this two conjetures about this
sequences: :

Numbers n equal to the sum of its proper divisors greater than cube root of
n.

18, 196, 222, 246, 258, 282, 318, 354, 366, 402, 426, 438, 474, 498, 534,
582, 606, 618, 642, 654, 678, 762, 786, 822, 834, 894, 906, 942, 978, 1002,
1038, 1074, 1086, 1146, 1158, 1182, 1194, 1266, 1338, 1362, 1374, 1398,
1434, 1446, 1506, 1542, 1578, 1614, 1626, 1662, 1686, 1698, 1758, 1842,
1866, 1878, 1902, 1986, 2022, 2082, 2094, 2118, 2154, 2202, 2238, 2274,
2298, 2334, 2382, 2406, 2454, 2514, 2526, 2586, 2598, 2634, 2658, 2694,
2742, 2766, 2778, 2802, 2874, 2922, 2946, 2994, 3018, 3054, 3126, 3138,
3246, 3282, 3342, 3378, 3414, 3426, 3462, 3522, 3558, 3594, 3606, 3642,
3678, 3702, 3714, 3786, 3846, 3858, 3882, 3918, 3954, 3966, 4038, 4062,
4098, 4146, 4206, 4254, 4314, 4362, 4398, 4434, 4458, 4506, 4542, 4566,
4614, 4638, 4722, 4782, 4854, 4866, 4926, 4938, 4962, 4974

is  407715 the only odd term is this sequence ?


Numbers n equal to the sum of its proper divisors greater than n^ 1/4

18, 1338, 1362, 1374, 1398, 1434, 1446, 1506, 1542, 1578, 1614, 1626, 1662,
1686, 1698, 1758, 1842, 1866, 1878, 1902, 1986, 2022, 2082, 2094, 2118,
2154, 2202, 2238, 2274, 2298, 2334, 2382, 2406, 2454, 2514, 2526, 2586,
2598, 2634, 2658, 2694, 2742, 2766, 2778, 2802, 2874, 2922, 2946, 2994,
3018, 3054, 3126, 3138, 3246, 3282, 3342, 3378, 3414, 3426, 3462, 3522,
3558, 3594, 3606, 3642, 3678, 3702, 3714, 3786, 3846, 3858, 3882, 3918,
3954, 3966, 4038, 4062, 4098, 4146, 4206, 4254, 4314, 4362, 4398, 4434,
4458, 4506, 4542, 4566, 4614, 4638, 4722, 4782, 4854, 4866, 4926, 4938,
4962, 4974

is  8415 the only odd term in this sequence ?

Best


2012/4/15 Jens Voß <jens at voss-ahrensburg.de>

> Am 15.04.2012 00:38, schrieb Psychedelic Geometry:
>
>> [...]
>>
>>
>> The abundances of the elements of A<10^6 are {12, 56, 810, 992} , (the
>> value 810 only happens for 34155)
>>
>>
> Hello,
>
> one obvious construction of elements of A is the following:
>
> If n is perfect and q a prime > n, then nq is in A (and its abundance is
> 2n). This accounts for the abundances 12, 56 and 992.
>
> A similar construction takes into consideration that all known perfect
> numbers have the form n = 2^(p-1) * (2^p - 1):
>
> If 2^(p-1) * (2^p - 1) is perfect, then 2^(p-1) * (2^p - 1)^3 lies in A.
> This accounts for the numbers 54 and 1372 in A.
>
> I have not had the time to check whether there are any exceptions (besides
> 34155) to these patterns - if not, this would really make 34155 very
> special!
>
> Regards,
> Jens
>



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