mersenneforum.org The natural progression of (even perfect numbers)*3
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2018-04-02, 23:56   #1
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

11100001101012 Posts
The natural progression of (even perfect numbers)*3

Quote:
 Originally Posted by 10metreh 2σ(a)-a
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.)

2018-04-03, 00:12   #2
10metreh

Nov 2008

232210 Posts

Quote:
 Originally Posted by Dubslow Is there an OEIS entry for this sequence (starting with f(0)=3)?
Yes: oeis.org/A146556

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

Last fiddled with by 10metreh on 2018-04-03 at 00:12

2018-04-03, 00:26   #3
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

3·29·83 Posts

Quote:
 Originally Posted by 10metreh Yes: oeis.org/A146556
Looks like the "Formula" is wrong? The "+ a(n-1)" at the beginning isn't right.

Quote:
 Originally Posted by 10metreh Seems not - actually g(3) = 30, and g(4) = 492. The only OEIS match for these terms is oeis.org/A143414 which is clearly unrelated.
Oops, yes I had an off-by-one in my local recreation.

2018-04-03, 12:25   #4
10metreh

Nov 2008

1001000100102 Posts

Quote:
 Originally Posted by Dubslow Looks like the "Formula" is wrong? The "+ a(n-1)" at the beginning isn't right.
It is correct - look carefully at the brackets.

2018-05-09, 12:37   #5
garambois

Oct 2011

5748 Posts

Quote:
 Originally Posted by Dubslow Originally Posted by 10metreh View Post 2σ(a)-a Is there an OEIS entry for these sequence (starting with f(0)=3) ?
And what's about odd perfect numbers, if they exist ?

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 :

http://www.aliquotes.com/parfait_2_z.txt

2018-05-15, 12:43   #6
Dubslow

"Bunslow the Bold"
Jun 2011
40<A<43 -89<O<-88

722110 Posts

Quote:
 Originally Posted by garambois Curiously, we do not find the prime number 3 in the decomposition into prime numbers of these terms.
Really? I'd even say that's more than a bit curious. Has anyone looked at this sequence before?

 2018-05-15, 15:59 #7 LaurV Romulan Interpreter     Jun 2011 Thailand 2·19·241 Posts 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 formulae for sigma 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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