Home Prime

Author

Eric W. Weisstein
April 14, 2011

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PrimeNumbers/HomePrime.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/HomePrime.html.

©2011 Wolfram Research, Inc. except for portions noted otherwise

Code

HomePrimeStep[n_]:=FromDigits[Flatten[IntegerDigits/@Flatten[Table[#1,{#2}]&@@@FactorInteger[n]]]]
HomePrime[n_Integer?Positive]:=NestWhile[HomePrimeStep,n,!PrimeQ[#]&]
HomePrimeList[n_Integer?Positive]:=NestWhileList[HomePrimeStep,n,!PrimeQ[#]&]

Open as of 4/14/2011

l={49,146,242,312,320,322,326,328,336,352,363,495,548,556,576,596,663,665,670,712,714,715,768,782,845,858,861,925,978,992};

http://www.rieselprime.de/Others/HomePrime10.htm

Enumeration

HomePrime/@Range[2,30]

TableForm[Table[{n,Length[h=HomePrimeList[n]]-1,h},{n,2,20}],TableDepth->2]

Search

60 Seconds

Do[Print[{n,TimeConstrained[HomePrime[n],60]//Timing}],{n,2,1000}]

homes={{2,{0.0000709999999948252`,2}},{3,{0.000021999999997746045`,3}},{4,{0.00010800000000088517`,211}},{5,{0.000019999999999242846`,5}},{6,{0.0000630000000008124`,23}},{7,{0.000017999999997186933`,7}},{8,{0.0027520000000045286`,3331113965338635107}},{9,{0.00009100000000117348`,311}},{10,{0.0001069999999963045`,773}},{11,{0.000019999999999242846`,11}},{12,{0.00004500000000007276`,223}},{13,{0.000019000000001767603`,13}},{14,{0.00014000000000535806`,13367}},{15,{0.00010800000000088517`,1129}},{16,{0.0001370000000022742`,31636373}},{17,{0.00002000000000279556`,17}},{18,{0.000044999999996520046`,233}},{19,{0.000018000000000739647`,19}},{20,{0.0017509999999987258`,3318308475676071413}},{21,{0.00005499999999969418`,37}},{22,{0.00004500000000007276`,211}},{23,{0.000019000000001767603`,23}},{24,{0.00007700000000099294`,331319}},{25,{0.00008500000000211116`,773}},{26,{0.00010899999999836041`,3251}},{27,{0.00010999999999938836`,13367}},{28,{0.00004500000000007276`,227}},{29,{0.000018000000000739647`,29}},{30,{0.00007000000000090267`,547}},{31,{0.000018000000000739647`,31}},{32,{0.00007200000000295859`,241271}},{33,{0.00004299999999801685`,311}},{34,{0.0001390000000043301`,31397}},{35,{0.00008600000000313912`,1129}},{36,{0.00007400000000146179`,71129}},{37,{0.00001899999999821489`,37}},{38,{0.00006500000000286832`,373}},{39,{0.0000439999999990448`,313}},{40,{0.0005769999999998277`,3314192745739}},{41,{0.000025000000004382628`,41}},{42,{0.00007199999999940587`,379}},{43,{0.000018000000000739647`,43}},{44,{0.0003829999999993561`,22815088913}},{45,{0.00017300000000020077`,3411949}},{46,{0.00004500000000007276`,223}},{47,{0.000018000000000739647`,47}},{48,{0.00790699999999589`,6161791591356884791277}},{49,{37.967324000000005`,$Aborted}},{50,{0.00019199999999841566`,3517}},{51,{0.00006399999999473494`,317}},{52,{0.000049000000004184585`,2213}},{53,{0.00001900000000887303`,53}},{54,{0.00004800000000670934`,2333}},{55,{0.00006600000000389628`,773}},{56,{0.0001949999999908414`,37463}},{57,{0.00009799999999415832`,1129}},{58,{0.000045999999997548`,229}},{59,{0.000020000000006348273`,59}},{60,{0.00007999999999697138`,35149}},{61,{0.000018999999994662176`,61}},{62,{0.00009399999999004649`,31237}},{63,{0.000046999999995023245`,337}},{64,{0.0005869999999958964`,1272505013723}},{65,{0.16872899999999902`,1381321118321175157763339900357651}},{66,{0.00009099999999762076`,2311}},{67,{0.00002599999999119973`,67}},{68,{0.00007200000000295859`,3739}},{69,{0.00009099999999762076`,33191}},{70,{0.000049000000004184585`,257}},{71,{0.00001900000000887303`,71}},{72,{0.0001010000000007949`,1119179}},{73,{0.00001900000000887303`,73}},{74,{0.00006499999999221018`,379}},{75,{0.00006600000000389628`,571}},{76,{0.000091999999995096`,333271}},{77,{37.292806`,$Aborted}},{78,{0.00040300000000570435`,3129706267}},{79,{0.000021000000003823516`,79}},{80,{1.3824040000000082`,313169138727147145210044974146858220729781791489}},{81,{0.0005220000000036862`,193089459713411}},{82,{0.000055999999986511284`,241}},{83,{0.000018999999994662176`,83}},{84,{0.00005099999999913507`,2237}},{85,{0.00009800000000836917`,3137}},{86,{0.011627999999987537`,6012903280474189529884459}},{87,{0.012394999999997935`,41431881512748629379008933}},{88,{0.00014199999999675583`,719167}},{89,{0.000021000000003823516`,89}},{90,{0.00010300000000995624`,71171}},{91,{0.00022599999999783904`,236122171}},{92,{0.00007700000000454565`,331319}},{93,{0.00004400000000259752`,331}},{94,{0.00006599999998968542`,1319}},{95,{0.00011099999998975818`,36389}},{96,{0.15081599999999185`,172929671097972226356946608292031596899264419}},{97,{0.000051000000013345925`,97}},{98,{0.00006300000001147055`,277}},{99,{0.0000780000000020209`,71143}},{100,{0.000102000000012481`,317047}},{101,{0.000020000000006348273`,101}},{102,{0.00007300000000043383`,7331}},{103,{0.00001900000000887303`,103}},{104,{0.0013670000000018945`,7241721710291471119}},{105,{0.05701100000000281`,3972297714730494749215180618316023}},{106,{0.00010400000002164234`,1123}},{107,{0.000020000000006348273`,107}},{108,{0.00007600000000707041`,23971}},{109,{0.000017999999982976078`,109}},{110,{0.00007799999998781004`,333331}},{111,{0.00004500000000007276`,337}},{112,{36.048699`,$Aborted}},{113,{0.00009600000001341868`,113}},{114,{0.00017300000001796434`,77711}},{115,{0.0000480000000209202`,523}},{116,{0.07001999999999953`,133756931160581891460350586392141}},{117,{0.00008599999998182284`,3313}},{118,{0.00011299999999891952`,3389}},{119,{0.00011799999998629573`,37199}},{120,{0.005027999999981603`,333331404995059281776857}},{121,{0.0001470000000267646`,127139}},{122,{0.00007099999999127249`,3329}},{123,{0.030337000000002945`,35091338111555586205509014201651}},{124,{0.000990999999999076`,291916529921668417}},{125,{0.00018900000000598993`,333131}},{126,{0.002087999999986323`,3331113965338635107}},{127,{0.000051000000013345925`,127}},{128,{0.00038000000000693035`,107234713298851}},{129,{0.5027550000000076`,3232690001463497545873217834379601755583}},{130,{0.00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todo=1+Flatten[Position[#[[2,2]]&/@homes,$Aborted]]

Length[todo]

3600 seconds

Print[{#,TimeConstrained[HomePrime[#],3600]//Timing}]&/@todo

home2={{49,{3600.096992`,$Aborted}},{77,{3600.05699`,$Aborted}},{112,{3600.0969920000016`,$Aborted}},{146,{3600.2290009999997`,$Aborted}},{234,{3600.1129930000016`,$Aborted}},{242,{3600.032987999997`,$Aborted}},{273,{3600.152995000002`,$Aborted}},{284,{3600.120993999997`,$Aborted}},{300,{3600.156995000005`,$Aborted}},{312,{3600.028987999998`,$Aborted}},{320,{3600.164996999996`,$Aborted}},{322,{3600.4890159999995`,$Aborted}},{326,{3597.5808349999998`,$Aborted}},{328,{3495.074428`,$Aborted}},{336,{3451.687717`,$Aborted}},{352,{3588.8882919999996`,$Aborted}},{360,{245.4273379999995`,11231805768648282840395848681269425026355001531739469354814326549}},{363,{3597.916855000003`,$Aborted}},{372,{169.37458600000173`,29928111574380178977502317300087951844711621861650093466161995947828637}},{407,{3422.6539029999985`,$Aborted}},{412,{1303.061436`,3891888106918927139261925557004544712052754712905085883093655994348874325979}},{460,{3598.120867999998`,$Aborted}},{495,{3598.256877`,$Aborted}},{542,{3281.753096999997`,$Aborted}},{548,{1800.1365020000012`,$Aborted}},{556,{1807.4409570000062`,$Aborted}},{558,{108.9548089999953`,4615821124702859186566305855547871821370125202080221901656295309097}},{576,{1818.7736670000013`,$Aborted}},{592,{3188.1832489999942`,$Aborted}},{596,{3480.5855230000016`,$Aborted}},{663,{2250.924674000009`,$Aborted}},{665,{3458.388134999994`,$Aborted}},{670,{3458.4281390000033`,$Aborted}},{693,{79.47296700000152`,32111190639144613312893877340615222196472964305217035784403643094979677}},{711,{3522.976171999995`,$Aborted}},{712,{3433.726594000007`,$Aborted}},{714,{3469.276815999998`,$Aborted}},{715,{3595.3926979999815`,$Aborted}},{744,{254.53190800000448`,47123843416687018015102064748607528900458921809886371641515295967153577335057}},{749,{3495.7544710000075`,$Aborted}},{762,{3551.1859359999944`,$Aborted}},{768,{3597.284814999992`,$Aborted}},{782,{3011.872230000008`,$Aborted}},{796,{3401.9446089999983`,$Aborted}},{800,{245.4233380000078`,3117112845952652977161953385713891508868301888869256285046791873}},{845,{3598.1488709999976`,$Aborted}},{847,{469.32933000000776`,13873988179327585669150690981650439625040022553353255166296622880114180860669}},{858,{3598.840913999986`,$Aborted}},{861,{3598.9929230000125`,$Aborted}},{864,{107.2987060000014`,131128468675119461523486098364608764905254769860682910413030160259009}},{866,{3599.21693699999`,$Aborted}},{867,{260.18826100000297`,43509631487712736035991113419372300726968487727153860769408998867864453287}},{896,{3598.0528639999975`,$Aborted}},{908,{3566.3668840000028`,$Aborted}},{925,{3599.304941999988`,$Aborted}},{964,{3596.960797000007`,$Aborted}},{969,{3598.8609140000044`,$Aborted}},{973,{3598.716905999987`,$Aborted}},{978,{3598.544894999999`,$Aborted}},{984,{182.9674350000132`,311829535912028769706176661305690093261695597427688657441600673261379568319}},{992,{3598.3528829999996`,$Aborted}}};

known=DeleteCases[{#[[1]],#[[2,2]]}&/@Union[homes,home2],{_,$Aborted}]

todo=Complement[Range[2,1000],First/@known]

Length[todo]

5 hours

Print[{#,TimeConstrained[HomePrime[#],3600]//Timing}]&/@todo

Steps

DeleteCases[Length/@HomePrimeList/@Range[2,20]-1,0]

HomePrime/@Select[Range[2,48],!PrimeQ[#]&]

TimeConstrained[For[i=1;n=49,
!PrimeQ[n],
Print[{i,Timing[n=HomePrimeStep[n]]}],
i++],3600]