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PrimeFan's Listing of Esoteric Integer Sequences

Main | Sylvester-type Sequences

Sylvester-type Sequences

SYLVS00000

Sequence

0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Description

Sylvester-type sequence starting with 0. Submitted by Anton Mravcek.

Mathematica Command

Sylvester0 = Table[0, {20}]; Do[Sylvester0[[m]] = (Times @@ Sylvester0[[Range[n]]]) + 1, {m, 2, 20}, {n, m - 1}]; Sylvester0

Thanks to Alonso Delarte for this program.

Comments

Numerically, for n = 0 forward, this sequence is the same as Sloane's A057427. This sequence is undefined for negative n, while A057427(n) = -1 for all negative n.


SYLVS00001

Sequence

1, 2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807

Description

Sylvester-type sequence starting with 1. Almost the same thing as the famous Sylvester's sequence starting with 2.


SYLVS00002

Sequence

2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, 12864938683278671740537145998360961546653259485195807, 165506647324519964198468195444439180017513152706377497841851388766535868639572 406808911988131737645185443

Description

Sylvester's sequence, starting with 2.

Comments

This sequence is Sloane's A000058.


SYLVS00003

Sequence

3, 4, 13, 157, 24493, 599882557, 359859081592975693, 129498558604939936868397356895854557, 16769876680757063368089314196389622249367851612542961252860614401811693, 281228763887799541044179637541757269144032017744154250062457419923642036907138172556157621940116222772834526664081092404683717311026229714557

Description

Sylvester-type sequence starting with 3.

Comments

This sequence is Sloane's A082732.


SYLVS00004

Sequence

4, 5, 21, 421, 176821, 31265489221, 977530816197201697621, 955566496615167328821993756200407115362021, 913107329453384594090655605142589591944556891901674138343716072975722193082773842421, 833764995101491832645293037100005838358401546554345116541504623640439618264388289392588552982541794273439064585807768109274107449211332094500263657984885302053765298821

Description

Sylvester-type sequence starting with 4.


SYLVS00005

Sequence

5, 6, 31, 931, 865831, 749662454731, 561993796032558961827631, 315837026779085485103718410756049100028793244531, 99753027484652761836825641888570160231510113732060091990981400322237526237557946030593168165431, 9950666492353889309155175255268577703999029170819543491300826665849917958384617974374977696339166746527626909734828276194928394332717632363876683892000351934782472846457509904310185015250331

Description

Sylvester-type sequence starting with 5.


SYLVS00006

Sequence

6, 7, 43, 1807, 3263443

Description

Sylvester-type sequence starting with 6.

Comments

Since 6 = 2 × 3, this sequence is exactly the same as Sylvester's sequence starting with 2, apart from the initial terms and subsequent offsets. We can generalize that any Sylvester-type sequence starting with a heteromecic number (a number of the form n(n + 1)) will be almost exactly the same as a Sylvester-type sequence starting with n (for example, the Sylvester-type sequence starting with 12 is very similar to the one starting with 3).


SYLVS00007

Sequence

7, 8, 57, 3193, 10192057, 103878015699193, 10790642145601683494645152057, 116437957914435303575899742229333045108448631998006179193, 13557798043283806950297045269968250387897834581711367551819275131055206893868524458302302046950954641412419952057, 183813887782470224479734703644752813086601972522989918614100638982767514395403028786664716951957824279710161721627297931077076048190274553251336631843362321234088265083946580752846621974171788413627931266342555287285758579193

Description

Sylvester-type sequence starting with 7.


SYLVS00008

Sequence

8, 9, 73, 5257, 27630793, 763460694178057, 582872231554839914154126117193, 339740038317718918529575265905277902175236102890836244082057, 115423293636125119010058921116641323017574325905706748548174179444923876042505937357640794991035512177751344518305269193, 13322536713811161396970968634181582989901582000358387856475164003950435593443166973669268075381910678496604518858624171814431767687033529695009935915794900290933475629143469137182356268444871735313240148529821039380791137691512709889602057

Description

Sylvester-type sequence starting with 8.


SYLVS00009

Sequence

9, 10, 91, 8191, 67084291, 4500302031888391, 20252718378218776104731448680491, 410172601707440572557971589875869064610540321970215293555320591, 168241563191450680898537024308131628447885486994777537422995633998657738457104605412468520116391629012196009150161991233268691, 28305223585102892620221589982451566984920480398257062507852816793038135939648182712734048008231784799302300061016919628041843326566167711604076865207421922526957426053994976039590613409530971557889572939390251984678779610004708652692061299852967584791

Description

Sylvester-type sequence starting with 9.


SYLVS00010

Sequence

10, 11, 111, 12211, 149096311, 22229709804712411, 494159998001727075769152612720511, 244194103625066907517263589918036880566782292998362610615987380611, 59630760245249914983032917582385072916976081620545731185395218096116199570911759967186110579865451606960736978164799881384991352711, 3555827567426477700894800760094842980241172443790856878511043749306776638335564658428071399148038697033377662771091994527731952262883955953931050592717248556233739954278546051597702451479718681056568614058247327472620453818138450548942977111657196562860615696811

Description

Sylvester-type sequence starting with 10.