Sorry about this, which turns out to be an unintentional hoax. Fred Helenius observes that I managed accidentally to insert the 19-digit prime into the term a(103), which is only 9311823293177912868609791130160292081658870701 Note that the offset in A005178 is ``wrong''. This is a ``divisibility sequence'', i.e., with the ``right'' offset a(m) divides a(n) just if m divides n, as in the Fibonacci sequence. The way to put it ``right'' is to define the n-th member to be the number of domino tilings of a 4 by n-1 rectangle [# of matchings of the graph P4 x P(n-1)] giving a(2)=1, a(1)=1 (the empty matching of the empty graph), a(3)=5: --- | | --- --- | | --- | | --- | | --- --- and | | --- --- --- a(n) = a(n-1) + 5a(n-2) + a(n-3) - a(n-4) with a(0) = 0 and a(-n) = a(n). Hugh Williams & I may have a paper within a finite time about such sequences (see also A003757) in which some primes have two ranks of apparition. R. On Sat, 20 Dec 2008, Richard Guy wrote:
Funsters & fansters might be amused by this somewhat unlikely coincidence. I had calculated a couple of hundred terms of a fourth order recurring sequence [it's A005178 in OEIS, if you want details] and was looking for the ranks of apparition of various primes. A 19-digit prime factor of the 53rd term turned up as a substring of the 103rd term!
Have a prime time in 7^2 x 41. R.
---------- Forwarded message ---------- Date: Wed, 17 Dec 2008 09:24:06 -0700 (MST) From: Richard Guy <rkg@cpsc.ucalgary.ca> To: Hugh Cowie Williams <williams@math.ucalgary.ca> Subject: You wouldn't believe
Hugh, before sending the final (??) version of the quadric file, I decided I'd do a search for larger primes just to see if there were any double ranks. While searching with
3140540902719737029
which is a factor of a(53), I discovered that it's a substring of a(103):
93118232931779128686097911301602920314054090271973702981658870701 ^^^^^^^^^^^^^^^^^^^ R.