Fwd: Prime Hyperfibonacci numbers

[Maximilian Hasler]

---------- Forwarded message ----------
From: Maximilian Hasler [using From: address not on SeqFan list...]
Date: Sat, Apr 5, 2008 at 10:49 AM
Subject: Re: Prime Hyperfibonacci numbers
To: Jonathan Post <jvospost3 at gmail.com>
Cc: Sequence Fans <seqfan at ext.jussieu.fr>

 It is not difficult to code the partial sums excluding the n<=2 and
k=0 columns/row:

 partsumfib(N,s=[],P=[])={
 for( n=1+#s,N, s=concat(s,n+1);
  forstep( i=n,1,-1,
  isprime( s[i]+=
   if( i>1, s[i-1], fibonacci(n+2) )
  ) & P=setunion(P,[s[i]])
  ); print(s);
 );vecsort(eval(P))}

 This produces the first N antidiagonals of A136431, and in the end
lists all primes found so far:
 (10:20) gp > partsumfib(10)
 [4]
 [7, 7]
 [12, 14, 11]
 [20, 26, 25, 16]
 [33, 46, 51, 41, 22]
 [54, 79, 97, 92, 63, 29]
 [88, 133, 176, 189, 155, 92, 37]
 [143, 221, 309, 365, 344, 247, 129, 46]
 [232, 364, 530, 674, 709, 591, 376, 175, 56]
 [376, 596, 894, 1204, 1383, 1300, 967, 551, 231, 67]
 %50 = [7, 11, 29, 37, 41, 67, 79, 97, 709, 967]

 I think that the list of primes is a priori only complete up to
(n+1)(n+2)/2 or so. For N=50 one gets
 7, 11, 29, 37, 41, 67, 79, 97, 137, 191, 211, 277, 379, 631, 709, 821,
 947, 967, 991, 1129, 1327...

 It is a nice idea to have built this table, and it would be nice if
it was completed with references for the columns
 (eg n=3: A000124, n=4: A004006, ... except for initial terms).
 Maybe it can be shown that  for each column n>3 there is only a
finite number of primes.
 However, I don't know if it is very natural to list the union of all
these primes in one sequence.

 Maximilian



 On Sat, Apr 5, 2008 at 3:33 AM, Jonathan Post <jvospost3 at gmail.com> wrote:
 > A generalization of prime Fibonacci numbers (A005478) is the prime
 >  hyperfibonacci numbers (primes in A136431).
 >
 >  Referring to the array a(k,n) =  Apply partial sum operator k times to
 >  Fibonacci numbers, we see that every prime occurs in the n=2 column
 >  (as it contains every positive integer).  So excluding n=2 and k=0
 >  ...


>  7, 11, 29, 37, 41, 79, 97, 709, 967, 2267, 2683, 8273, 10093, 14323,
 >  16703, 17519, 127921, 226007, 514201, 23945893
 >
 >  Does someone want to code iterated application of the partial sum
 >  operator k times to Fibonacci numbers, and test for primes?  I'd be
 >  pleased to be co-author with whomsoever wants to do that coding and
 >  primality testing.
 >
 >  Best,
 >
 >  Jonathan Vos Post
 >