[seqfan] Re: Integers with a peculiar divisibility property

[Georg.Fischer]

Hi Allan,

though I'm a PARI beginner, I get - like Sean an Harvey -
in a few seconds (for n^1):

(11:19) gp > for(n=1, 10000, if(Mod(sigma(sigma(n^4)),  (n)) == 0, 
print1(n", ")))
1, 2, 4, 8, 16, 19, 21, 25, 32, 38, 42, 50, 57, 64, 76, 84, 96, 100, 
114, 148, 150, 152, 168, 200, 228, 256, 304, 336, 399, 400, 416, 444, 
448, 456, 461, 475, 525, 608, 640, 672, 768, 798, 800, 912, 922, 950, 
1036, 1050, 1053, 1088, 1152, 1197, 1216, 1248, 1280, 1332, 1344, 1383, 
1408, 1425, 1539, 1596, 1664, 1792, 1824, 1844, 1900, 1911, 1920, 2016, 
2048, 2075, 2100, 2106, 2205, 2220, 2304, 2394, 2400, 2432, 2483, 2592, 
2688, 2752, 2766, 2793, 2812, 2850, 2944, 3078, 3108, 3136, 3150, 3192, 
3200, 3227, 3269, 3392, 3411, 3456, 3648, 3688, 3700, 3744, 3800, 3822, 
3840, 3996, 4050, 4149, 4150, 4200, 4212, 4224, 4410, 4704, 4788, 4864, 
4966, 4992, 5376, 5408, 5472, 5532, 5586, 5673, 5700, 5733, 5760, 5779, 
5993, 6156, 6384, 6400, 6454, 6538, 6822, 6912, 7040, 7252, 7296, 7376, 
7449, 7471, 7600, 7616, 7644, 7904, 8064, 8256, 8298, 8300, 8320, 8400, 
8424, 8436, 8512, 8550, 8736, 8759, 8820, 8832, 8960, 9324, 9576, 9600, 
9681, 9728, 9807, 9932, 9975,

I don't see why 21 is not primitive?

n^2 took 9 minutes:
(11:22) gp > for(n=1, 100000, if(Mod(sigma(sigma(n^4)),  (n^2)) == 0, 
print1(n", ")))
1, 2, 4, 8, 16, 19, 38, 76, 152, 304, 608, 1216, 1824, 3744, 3840, 4864, 
6400, 7904, 11520, 14592, 19200, 21888, 23712, 24320, 25536, 32768, 
33696, 34560, 43776, 71136, 72960, 80640,
(11:31) gp >

Both sequences might make their way into the OEIS.

(11:31) gp > for(n=1, 100000, if(Mod(sigma(sigma(n^4)),  (n^3)) == 0, 
print1(n", ")))
1, 2, 8, 16, 608, 32768,
(11:38) gp >

Best regards - Georg

Am 07.01.2019 um 04:39 schrieb Sean A. Irvine:
> Hi Allan,
> 
> I'm not seeing some of the values you report, for example, I get
> 
> sigma(sigma(21^4))/21^2 = 17746/21.
> 
> Did you mean divisible by n rather than n^2?
> 
> For sigma(sigma(n^4)) divisible by n^2, I get
> 
> 1, 2, 4, 8, 16, 19, 38, 76, 152, 304, 608, 1216, 1824, 3744, 3840, 4864,
> 6400, ...
> 
> (still not in the OEIS though)
> 
> Sean.
> 
> 
> 
> 
> On Mon, 7 Jan 2019 at 14:02, Allan Wechsler <acwacw at gmail.com> wrote:
> 
>> Let sigma be the familiar sum-of-divisors function.
>>
>> While hunting multiply perfect numbers, I learned that
>>
>> sigma(19^4) = 151 * 911,
>>
>> and sigma(151) and sigma(911) are both divisible by 19. This prompted me to
>> investigate which numbers n have sigma(sigma(n^4)) divisible by n^2.
>>
>> I wrote a Haskell one-liner to list them -- but it computes sigma by brute
>> force, so it's very very slow. So far it has found:
>>
>> 1,2,4,8,16,19,21,25,32,38,42,50,57,64 ... which is not in OEIS.
>>
>> The "primitive" elements (not products of smaller elements) are 2, 19, 25,
>> ... and even this tiny three-element sequence is not in OEIS. Can anyone
>> whip up a faster program to look for bigger primitives?
>>
>> --
>> Seqfan Mailing list - http://list.seqfan.eu/
>>
> 
> --
> Seqfan Mailing list - http://list.seqfan.eu/
> 

-- 
Dr. Georg Fischer, Rotteckring 19, D-79341 Kenzingen
Tel. (07644) 913016, +49 175 160 7788, www.punctum.com