Prime Hyperfibonacci numbers

[Joshua Zucker]

On Sat, Apr 5, 2008 at 12:33 AM, Jonathan Post <jvospost3 at gmail.com> wrote:
 >  k=1: primes in A000071 = {A000071(4) = 7}, no more through n = 36.

 No more through n = 1000.


 >  k=2: primes in A001924 = {A001924(3) = 7, A001924(7) = 79,
A001924(25) = 514201}

 7 79 514201 14930317 956722025983 5527939700884681 4660046610375530219
 51680708854858322977 110560307156090817237632754212191
 6867260041627791953052057353082063289320595801
 1032438788598817315437189032298124708705297018004363191791263255924514005104084376841381572199036650610593743671014987440620683963
 64128128989018176585252040302126085488071926335079419810451272680747571571160723032344138301571590200724086190900349998530438936403887378197501



 >  k=3: primes in A014162 = {A014162(3) = 11, A014162(6) = 97,
 >  A014162(16) = 17519}, no more through n = 30.

 11 97 17519 47725408630716931427100836606467944347137848743555937753109926287696392817941313769896270467233014298574018232891991586395808269488787648022660796180183028113926753680531994000177681



 >  k=4: primes in A014166 = {A014166(4) = 41, A014166(13) = 10093,
 >  A014166(14) = 16703}

 41 10093 16703 3520457 591286703533 6557470285501 19740274219868101499
 3686839203203908875430996692179542915310607843499662532720252170412169301591070559195321429910935928304254470448268332995152315903729



 >  k=5: primes in A053739 = {A053739(7) = 709, A053739(10) = 8273,
 >  A053739(11) = 14323}, no more through n = 27.

 709 8273 14323 4660046610373296841 298611126818977061133263
 11463113765491467695340528626391998567
 113796925398360272257523782552224175572745930353730513145086634176691092536145985470146129334641866902783673042322088625863396052888690096969577173696370562180400527049497109023054114771394568040040371085351113



 >  k=6: primes in A053295 = {A053295(3) = 29, A053295(8) = 2683,
 >  23945893(24) = 23945893}, no more through n = 27.

 29 2683 23945893 1835540197 4052735290427 27777884012083
 2111485065762857 81055900096023504196858896749
 2353412818241252672952148474817 468340976726457153752542674767203
 100694286476841731898333719576864360661213863366440731029849
 8531073606282249384383143963212896619394786170594625964346924608340860622589
 99080696264900721472208515956748759187919804840608734007367661059706052043279378932885908102386332223129673921



 >  k=7: primes in A053296 = {A053296(3) = 37, A053296(6) = 967,
 >  A053296(7) = 2267, A053296(12) = 127921, A053296(13) = 226007}, no
 >  more through n = 27.

 37 967 2267 127921 226007 62048869 1131463777 7540113804271826929
 11636390653418416980850249915594581159038678944868584489700931605802057426181849
 317445252231252689168194927677674100517887360125535240613651188143594986671799019825105289947239151



 >  k=8: primes in A053308, none through n = 27.

 27777538280521 1409869790947669143312035590804646728957
 690168906931029935139391829792095612517948949938312818533113
 3412522850068441236875759409983150176537865248321769437824938166162045767256336787279692556575755320901402251
 911461723979229137398188463207134083118652096691179548565493854813244058514158710983136691744844421148519387154025168747083889757741096637632733083868351676999179657
 247409904738736056360531950204272194467455305702693665155191284116520387970821525665002988000629206259302085539829969039201607330980012683779731383105758142727782176348381



 >
 >  k=9: primes in A053309, none through n = 26.

 1033628323428189498226451492123369099
 17690617586682990180447203622188161240682337031027142006892310369574875779255058883026493511364861
 196191955446197556957565929345772792668594307949581132632670453793550007197467504974396227393748639
 5104867695441585599646129198394874763979069372338122901969709030921057521006570452530842908208673336523400258809842467982145963437107797526232075655133690089193142043267298037061

 k=10:
 67 5972304273877744135569337875802249660927

 k=11:
 79 4478413 19008291293 61305228407581679
 17978720198565577104981084195543091067883412951
 22334640661774067356412331900038009953045351020683823507194721380325737329411
 432995556774426913953123610518785740738997352293147773391602699511793756235520810476170490422053365763
 6963043279151935385260634152489320944063572302406363313807306655433558205159444721340676247903232264078005633069442343500128939954620940916151759620589495754171588874454552652561061

 k = 12:
 6763 1982269 37886753582095837 2791715456569622316696636389
 271964099255182923543922814194421841992005148669513
 196191955446197556957565929345772792668594307949581132632670453793550007197463060594255375221298723
 12186120053215401266088089750036680129447500839931526579142040485499190586947405129163013698045888837789667669811
 41505733030349875599942168299716793497054143143751575205826222961703655325391028507041891275315653078351800637872570251490507155509441011827371957348902611511529241338126738309



 >
 >  So, without going past k=9 or those values already in the existing
 >  OEIS sequences other than A000045 (A000071, A001924, A014162, A014166,
 >  A053739, A053295, A053296, A053308, A053309, A123736) 20 of the the
 >  nontrivial hyperfibonacci primes include:
 >
 >  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.

 I think I did it ok here.  I didn't test for prime, only pseudoprime,
 so if you put any of the larger numbers into the OEIS it would be a
 good idea to run them through the Alpertron or some such first.  In
 each case above I checked for primes through n = 1000 (with my
 definition -- the first k+1 terms are all 0s.  Maybe the usual
 definition means I only checked through n = 1000-k-1 then, or
 something)

 Just for fun, here's all the primes less than 10^6 that ever occur as
 hyperfib numbers, sorted by k and then within k by n, with
 multiplicity (so if a number occurs as two different hyperfib it will
 appear here twice):
 7 7 79 514201 11 97 17519 41 10093 16703 709 8273 14323 29 2683 37 967
 2267 127921 226007 67 79 6763 137 59281 80789 191 211 243157 277 379
 631 821 947 991 1129 1327 1597 1831 2017 2081 2347 2557 2851 2927 3571
 3917 4561 4657 4951 5051 5779 6217 6329 8647 8779 9181 9871 11027
 12721 13367 14029 14197 14879 15401 16111 17021 17579 20101 20707
 21529 21737 22367 24091 24977 26107 27967 28921 31627 33931 34981
 36857 37951 39341 40471 41617 42487 43661 44851 47279 50087 51361
 53629 58997 59341 60727 62129 64621 66067 66431 69379 70501 70877
 75079 77029 78607 79801 83437 87991 88411 91807 93097 93529 95267
 96581 97021 98347 100129 106031 107881 109279 113051 117371 122761
 126757 127261 129287 130817 133387 135461 145531 150427 152077 152629
 154291 156521 157081 163307 165601 167911 168491 177907 182107 182711
 185137 189421 194377 196879 197507 200029 202567 204481 210277 212227
 215497 218131 222779 225457 226129 230861 234271 236329 239087 239779
 241861 248161 250279 255971 256687 271217 274171 282377 289181 292231
 294529 297607 310867 319601 320401 322807 329267 333337 342379 346529
 352381 356591 359129 363379 365941 372817 377147 380629 387641 394717
 411779 426427 427351 433847 438517 442271 445097 446041 448879 468029
 471907 484621 488567 491537 495511 500501 512579 515621 519691 523777
 524801 531997 548629 557041 565517 575129 579427 595687 596779 605551
 608857 613279 614387 617717 632251 641279 653797 658379 668747 673381
 682697 705079 706267 730237 733867 738721 743591 744811 749701 769421
 773147 779377 784379 799481 809629 818561 828829 830117 839161 840457
 844351 893117 903841 914629 930931 941879 947377 957037 968137 990529
 996167

 My program runs fast (it's just a few lines of Scheme -- I have a
 prime? function from a library) so if there's any more you want from
 any of this, let me know.

 Enjoy,
 --Joshua Zucker