Number-Divisors Almost = ln(m) + 2c-1
Leroy Quet
qq-quet at mindspring.com
Wed Dec 3 00:22:39 CET 2003
>On Dec 2, 2003, at 1:44 PM, <all at abouthugo.de> wrote:
>
>> I checked from 2..128 and found records for
>> 3: 2 - ln(3) +1 -2*0.5772156649=0.746956382
>> 5: 2 - ln(5) +1 -2*0.5772156649=0.236130758
>> 7: 2 - ln(7) +1 -2*0.5772156649=-0.100341479
>> 46: 4 - ln(46) +1 -2*0.5772156649=0.01692727
>
>I get: 1, 3, 5, 7, 46, 2514, 2522, 2526, 2534, 2536, 2542, 2546, 2553,
>2555, 18873, 139454, 139475, 7614005, 7614010, 7614015, 7614022,
>7614030, 7614033, 7614034, 7614056, 7614062, 7614066, 7614069, 7614079,
>7614082, 7614086, 7614087, 7614088, 7614104, 7614113, 7614114, 7614123,
>7614127, 7614130, 7614135, 7614136, 7614139, 7614142, 7614145, 7614158,
>7614165, 7614166, 7614168, 7614174, 7614184, 7614190, 7614195, 7614219,
>7614232, 7614245, 7614249, 7614255, 7614257, 7614265, 7614266, 7614269,
>7614280, 7614298, 7614310, 7614312, 7614318, 7614321, 7614326, 7614327,
>7614330, 7614344, 7614345, 7614357, 7614365, 7614385, 7614386, 7614392,
>7614395, 7614397, 7614398, 7614406, 7614408, 7614411, 7614415, 7614422,
>7614435, 7614456, 7614459, 7614474, 7614481, 7614483, 7614489, 7614490,
>7614498, 7614526, ...
>
>For the last one, the value is 3.733908 * 10^-7.
>
>> Should be very easy to extend this with Maple's tau,
>> Mathematica's DivisorSigma or Pari's numdiv ...?
>
>Mathematica's DivisorSigma does sum of (powers of) divisors. For
>number-of-divisors, I use Length[Divisors].
>
At first I thought the above sequence was unusual when I noticed it
contained large jumps in the integers followed by long stretches with
very little increase.
But I soon realized that this was not unexpected at all.
(For the stretch from 7614005 to 7614526, for example, are simply the
integers in that range with 16 divisors, because 16 approximates ln(a(n))
+2c -1 for a relatively large number of integers a(n). {duh!})
Still, this suggests other EIS sequence(s), such as the number of
integers m, where
d(m) approximately equals ln(m) +2c -1, whatever 'approximately' means
(exactly).
Or how about the sequence of m's, where each successive m is better
approximated by
exp(d(m)-2c+1), or some other related sequence?
thanks,
Leroy Quet
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