## 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);