## Monday, October 25, 2010

### The "phi-bonacci" sequence - an update

More data on the "phi-bonacci" sequence introduced previously here, after computing the first 500 terms:
• a plot of the sequence of quotients X(n+1)/X(n) for n = 1,2,...,499:

X(n) is a multiple of 4 for n between 13 and 500.
Here is the updated raw data (last previously calculated term marked in red):

## Friday, October 22, 2010

### GPF stability?...

A visual on the behavior of the same recurrence as before, only with a different initial condition (an 8-digit prime, picked at random)
L[1] = 11631013
L[N] = P(26390*L[N-1] + 1103)
The limit cycle is the same. The choice
L[1] = 7654237825827857857221111238572389123865443346789678979
leads, again to the same limit cycle. This raises an interesting question of "stability" (that is, assuming ultimate periodicity holds, are there finitely many - if not a single one - limit cycles?*) Note that in this particular case, the limit cycle is not unique: for example, the choice
L[1] = 2250957258971258907129712971234237484736596896123596812363
leads to a different limit cycle, of period 18:

## Thursday, October 21, 2010

### A recurrence with primes inspired by a formula of Ramanujan

Inspired by the linear component appearing in the numerators of the terms of the Ramanujan's formula for pi, I looked into the recurrent sequence of primes defined as follows:
L[1] = 2
L[N] = P(26390*L[N-1] + 1103)
where P is the greatest prime factor function. The prime sequence (L[N]) turns out to be ultimately periodic, with period
(1459, 30011, 15529243, 409816723873, 292299009270529, 701251895877205583, 15696384675317604187, 451826639233, 109391789076697, 151939437564949207, 74396630251, 29303389139179, 26646743, 2111734381, 55728670315693, 70865191, 18516162293, 487831, 12873861193, 1132987, 1921687, 330167, 968123137, 2901559, 14505047, 2091738751, 675347, 5940802811, 57192517, 137210047703, 938359, 24763295113, 6250999, 2795997707, 6922448587, 30040009, 46632696389, 3290305727, 282788861, 24151449977, 212452254964711, 5606615008518724393, 4871183188935589, 733134419023, 19347417318018073, 29451120121, 218661017, 242650193, 36178184149, 16311737023, 6797093683, 179375302295473, 29858734728029, 1862817989297131, 4993054962517, 15723597697, 311287129201, 13424039, 6946282163, 252731, 741063577, 106787, 26293, 10356319, 23227, 612961633, 511873, 4640443, 346915841, 3511741099, 33101671, 3527213, 6883, 20719, 433, 57427, 1529263, 510851287, 4456663, 117611337673, 999421747, 33078701, 2389547, 2335560979, 4285298911, 113089038262393, 30741542832733, 1801335599663, 688945601088517, 4790671, 7790117, 769966999, 239280127, 8821951, 756227, 34513, 4289, 37729271, 914302537, 11594639093, 39465611, 11110373, 478571, 11549, 859, 22670113, 176369, 1551459671, 40453493, 6714262147, 14927447, 43770591937, 70138194257, 2603300909203, 51190093, 942712181, 8292724819231, 24011017, 168928483, 6954793553, 270216913, 16301459, 1158523, 30573423073, 21129377, 5023461803, 1979, 17408971, 70381, 1798021, 88289, 467017, 540149, 39929, 4231837, 223803967, 5906186690233, 155864266755249973, 144041128039, 604427630617, 8896176894581, 1577342687387, 17151245784979, 452621376265596913, 878711663, 19359229, 2829679, 12128509, 8627, 41177, 65371, 1663589, 4878012757, 207798167, 609310403137, 1409652983, 1377805267499, 526960594337677, 86463252140809, 8419797874523803, 15369684199, 10962323405749, 1278583667, 132525119, 129531033019, 1221680363, 7825861, 206524472893, 5450180839647373, 24708859707660913, 652066807685171495173, 1941559636106473627171, 193285868819, 100015962316363, 1111967795377, 29344830120000133, 81694832084891, 718642206240091531, 18964967822676015504193, 4407663528935561, 857965267921, 42742695533, 20327255503, 11413537717559, 250324962137, 1275058048793, 11216260635882791, 857911268779, 13341354380129, 39119815787956157, 209783083, 325657385969, 72031783, 2035949, 163601, 72953, 7247, 21249937, 560785838533, 1345376207171543, 1075893275977485481, 9099479420859497, 98635636088557, 320878911329, 2822664823324471, 110249906411, 14055531546799, 499608015161, 548195730743, 160630728871, 385367721355163, 32609594737, 1641473, 83465267, 122840243, 140682811, 161071, 4250664793, 153443, 449929097, 26904379, 116579, 405499, 7715299, 3579647, 33083, 2881391)
Here is a logarithmic plot of this sequence:

This special case illustrates a general conjecture on the ultimate periodicity of GPF sequences. For this, and related sequences and algebraic structures, see
• Greg Back and Mihai Caragiu, The Greatest Prime Factor and Recurrent Sequences, Fibonacci Quarterly (accepted for publication);
• Mihai Caragiu and Greg Back, The Greatest Prime Factor and Related Magmas, JP J.of Algebra, Number Theory and Appl. 15 (2), 127-136 (December 2009);
• Mihai Caragiu and Lisa Scheckelhoff, The Greatest Prime Factor and Related Sequences, JP J.of Algebra, Number Theory and Appl. 6(2), 403-409 (2006);