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• ## Hey factorers... a sequence of numbers that are never prime

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• A[0]=0 A[1]=1 A[n] = 3*a[n]-a[n-1]+2 this begins 0, 1, 5, 16, 45, 121, 320 and there is a page on it here: http://oeis.org/A004146 The parities repeat
Message 1 of 4 , Mar 30
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A[0]=0
A[1]=1
A[n] = 3*a[n]-a[n-1]+2
this begins
0, 1, 5, 16, 45, 121, 320
and there is a page on it here:
http://oeis.org/A004146

The parities repeat (even,odd,odd).
Strangely enough, every element of this sequence is either
I. a square, or
II. 5 times a square
(these two types alternate). That already seems rather peculiar.
The A[n] of type I's can be, and often are, primes^2, in fact that happens (for n<9999)
exactly when
n=5, 7, 11, 13, 17, 19, 31, 37, 41, 47, 53, 61, 71, 79, 113,
313, 353, 503, 613, 617, 863, 1097, 1361, 4787, 4793, 5851,
7741, 8467
which note are all themselves prime.
This coincides with http://oeis.org/A168033.
But the type II's apparently only exhibit a single case of being 5*prime^2, namely
45=5*3^2 where 3=prime
(based on me testing A[1,2,...,9999]).
This latter fact seemed really peculiar (but I finally understood why:)

There is a sense in which these A[j] can be regarded as being like Mersenne numbers
2^r-1 but (sort of) to an irrational base (the golden number 1.61803...) instead of 2. It is well known Mersennes can only be prime if r=prime. I can prove the analogue here.
It is agreed it is interesting to factor Mersenne numbers and tabulate the results. I think it is also interesting to factor these and tabulate results.
• It seems the correct formula is: A[0] = 0; A[1] = 1; A[n] = 3*A[n - 1] - A[n - 2] + 2; -- Mathematics is the queen of the sciences and number theory is the
Message 1 of 4 , Mar 30
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It seems the correct formula is:

A[0] = 0;
A[1] = 1;
A[n] = 3*A[n - 1] - A[n - 2] + 2;

--
"Mathematics is the queen of the sciences and number theory is the queen of
mathematics."
--Gauss

[Non-text portions of this message have been removed]
• ... F(n)=fibonacci(n); L(n)=if(n,F(2*n)/F(n),2); A(n)=if(n%2,L(n)^2,5*F(n)^2); print(vector(15,n,A(n))); [1, 5, 16, 45, 121, 320, 841, 2205, 5776, 15125,
Message 1 of 4 , Mar 31
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"WarrenS" <warren.wds@...> wrote:

F(n)=fibonacci(n);
L(n)=if(n,F(2*n)/F(n),2);
A(n)=if(n%2,L(n)^2,5*F(n)^2);
print(vector(15,n,A(n)));

[1, 5, 16, 45, 121, 320, 841, 2205, 5776, 15125, 39601, 103680, 271441, 710645, 1860496]

David
• ... http://mersennus.net/fibonacci/ provides complete factorizations of A(n) for 0
Message 1 of 4 , Mar 31
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> F(n)=fibonacci(n);
> L(n)=if(n,F(2*n)/F(n),2);
> A(n)=if(n%2,L(n)^2,5*F(n)^2);

http://mersennus.net/fibonacci/
provides complete factorizations of A(n) for 0 < n < 1159:
http://mersennus.net/fibonacci/lucholes.txt

David
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