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-   -   The natural progression of (even perfect numbers)*3 (https://www.mersenneforum.org/showthread.php?t=23217)

Dubslow 2018-04-02 23:56

The natural progression of (even perfect numbers)*3
 
[QUOTE=10metreh;484026]2σ(a)-a[/QUOTE]

Is there an OEIS entry for this sequence (starting with f(0)=3)?

Or, as a corollary, the secondary sequence g(n) where g(n) is the first index of f(n) which is divisible by the nth Mersenne prime?

Such that we know at least the first 873 terms of f(n); meanwhile g(1) = 0, g(2) = 2, g(3) = 29, with g(5) unknown as evidenced by this thread?

(Of course Mp | f(n) does not guarantee that the nth term loses the Mp driver, but it does mean that the sequence diverges from the f sequence.)

10metreh 2018-04-03 00:12

[QUOTE=Dubslow;484083]Is there an OEIS entry for this sequence (starting with f(0)=3)?[/quote]

Yes: [URL]oeis.org/A146556[/URL]

[quote]Or, as a corollary, the secondary sequence g(n) where g(n) is the first index of f(n) which is divisible by the nth Mersenne prime?

Such that we know at least the first 873 terms of f(n); meanwhile g(1) = 0, g(2) = 2, g(3) = 29, with g(5) unknown as evidenced by this thread?[/quote]

Seems not - actually g(3) = 30, and g(4) = 492. The only OEIS match for these terms is [url]oeis.org/A143414[/url] which is clearly unrelated.

Dubslow 2018-04-03 00:26

[QUOTE=10metreh;484085]Yes: [URL]oeis.org/A146556[/URL]

[/quote]

Looks like the "Formula" is wrong? The "+ a(n-1)" at the beginning isn't right.


[QUOTE=10metreh;484085]
Seems not - actually g(3) = 30, and g(4) = 492. The only OEIS match for these terms is [url]oeis.org/A143414[/url] which is clearly unrelated.[/QUOTE]

Oops, yes I had an off-by-one in my local recreation.

10metreh 2018-04-03 12:25

[QUOTE=Dubslow;484088]Looks like the "Formula" is wrong? The "+ a(n-1)" at the beginning isn't right.[/QUOTE]

It is correct - look carefully at the brackets.

garambois 2018-05-09 12:37

[QUOTE=Dubslow;484083]

Originally Posted by 10metreh View Post
2σ(a)-a

Is there an OEIS entry for these sequence (starting with f(0)=3) ?

[/QUOTE]

And what's about odd perfect numbers, if they exist ?

:smile:

Starting with f(0)=2, we have : 2, 4, 10, 26, 58, 122, 250, 686 ...
There is no OEIS entry for this sequence.

Just for fun, I calculated the first 348 terms of this sequence. Curiously, we do not find the prime number 3 in the decomposition into prime numbers of these terms.

You can see here :

[URL="http://www.aliquotes.com/parfait_2_z.txt"]http://www.aliquotes.com/parfait_2_z.txt[/URL]

Dubslow 2018-05-15 12:43

[QUOTE=garambois;487274] Curiously, we do not find the prime number 3 in the decomposition into prime numbers of these terms.[/QUOTE]

Really? I'd even say that's more than a bit curious. Has anyone looked at this sequence before?

LaurV 2018-05-15 15:59

As we start with a number n which is either 1 or 2 (mod 3) and 0 (mod 2), it means this is either 2 or 4 (mod 6). If its sigma is 0 (mod 3), then we have the next term t=2*sigma-n is either t=2*0-1=2 (mod 3) or t=2*0-2=1 (mod 3). So the only combinations that would result in 0 (mod 3) are either (A) n=4 (mod 6) and sigma(n)=2 (mod 3) or (B) n=2 (mod 6) and sigma(n)=1 (mod 3). We can expand some [URL="https://en.wikipedia.org/wiki/Divisor_function#Formulas_at_prime_powers"]formulae for sigma[/URL] and see if we can get this, considering that if n=1 (mod 3) than its prime factors which are 2 (mod 3) can be grouped two-by-two in those products for sigma, etc.


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