Smarandache-Wellin primes

[Ignacio Larrosa Cañestro]

Sunday, January 18, 2004 7:37 PM [GMT+1=CET],
N. J. A. Sloane <njas at research.att.com> escribió:

> There is a conflict between the Crandall-Pomerance book and the OEIS.
> Would someone please check?
>
> The book says the concatenation of the first 719 primes is a prime.
> This is the number  235711...5441 (since the 719-th prime is 5441).
>
> Yet in the OEIS we have:
>
> %S A019518
> 2,23,235,2357,235711,23571113,2357111317,235711131719,23571113171923,
> %T A019518 2357111317192329,235711131719232931,23571113171923293137,
> %U A019518
> 2357111317192329313741,235711131719232931374143,23571113171923293137414347
> %N A019518 Smarandache-Wellin numbers: a(n) = concatenation of first
> n primes. (fine)
>
> %I A069151
> %S A069151 2,23,2357
> %N A069151 Concatenations of consecutive primes, starting with 2,
> that are also prime. %C A069151 Prime Smarandacher-Wellin numbers.
> ...
> %E A069151 The next term has 355 digits and is too large to include.
> (presumably fine, but I did not check "355")

Yes it is 355 long:

23571113171923293137414347535961677173798389971011
03107109113127131137139149151157163167173179181191
19319719921122322722923323924125125726326927127728
12832933073113133173313373473493533593673733793833
89397401409419421431433439443449457461463467479487
49149950350952152354154755756356957157758759359960
16076136176196316416436476536596616736776836917017
09719 ~= 2.357111317·10^354
(128 primes)

As you see, the 128-th prime is 719. Would be that what Crandall-Pomerance
want to say?

The next ones are:

23571113171923293137414347535961677173798389971011
03107109113127131137139149151157163167173179181191
19319719921122322722923323924125125726326927127728
12832933073113133173313373473493533593673733793833
89397401409419421431433439443449457461463467479487
49149950350952152354154755756356957157758759359960
16076136176196316416436476536596616736776836917017
09719727733739743751757761769773787797809811821823
82782983985385785986387788188388790791191992993794
1947953967971977983991997100910131019102110311033

~= 2.357111317·10^498
(174 primes)

23571113171923293137414347535961677173798389971011
03107109113127131137139149151157163167173179181191
19319719921122322722923323924125125726326927127728
12832933073113133173313373473493533593673733793833
89397401409419421431433439443449457461463467479487
49149950350952152354154755756356957157758759359960
16076136176196316416436476536596616736776836917017
09719727733739743751757761769773787797809811821823
82782983985385785986387788188388790791191992993794
19479539679719779839919971009101310191021103110331
03910491051106110631069108710911093109711031109111
71123112911511153116311711181118711931201121312171
22312291231123712491259127712791283128912911297130
11303130713191321132713611367137313811399140914231
42714291433143914471451145314591471148114831487148
91493149915111523153115431549155315591567157115791
58315971601160716091613161916211627163716571663166
71669169316971699170917211723173317411747175317591
77717831787178918011811182318311847186118671871187
31877187918891901190719131931193319491951197319791
98719931997199920032011201720272029203920532063206
92081208320872089209921112113212921312137214121432
15321612179220322072213222122372239224322512267226
922732281228722932297 ~= 2.357111317·10^1170
(342 primes)

23571113171923293137414347535961677173798389971011
03107109113127131137139149151157163167173179181191
19319719921122322722923323924125125726326927127728
12832933073113133173313373473493533593673733793833
89397401409419421431433439443449457461463467479487
49149950350952152354154755756356957157758759359960
16076136176196316416436476536596616736776836917017
09719727733739743751757761769773787797809811821823
82782983985385785986387788188388790791191992993794
19479539679719779839919971009101310191021103110331
03910491051106110631069108710911093109711031109111
71123112911511153116311711181118711931201121312171
22312291231123712491259127712791283128912911297130
11303130713191321132713611367137313811399140914231
42714291433143914471451145314591471148114831487148
91493149915111523153115431549155315591567157115791
58315971601160716091613161916211627163716571663166
71669169316971699170917211723173317411747175317591
77717831787178918011811182318311847186118671871187
31877187918891901190719131931193319491951197319791
98719931997199920032011201720272029203920532063206
92081208320872089209921112113212921312137214121432
15321612179220322072213222122372239224322512267226
92273228122872293229723092311233323392341234723512
35723712377238123832389239323992411241724232437244
12447245924672473247725032521253125392543254925512
55725792591259326092617262126332647265726592663267
12677268326872689269326992707271127132719272927312
74127492753276727772789279127972801280328192833283
72843285128572861287928872897290329092917292729392
9532957296329692971299930013011301930233037
~= 2.357111317·10^1542

(435 primes)


And any other with less of 750 primes.

Saludos,

Ignacio Larrosa Cañestro
A Coruña (España)
ilarrosa at mundo-r.com

> %S A046035 1,2,4,128,174,342,435,1429
> %N A046035 Concatenation of first n primes is prime.
> ...
> %A A046035 Eric W. Weisstein (eric(AT)weisstein.com)
> %E A046035 Additional comments from Robert G. Wilson v
> (rgwv(AT)rgwv.com), Sep 10 2001 (surely 719 should be there?)
>
> %S A046284 2,3,7,719,1033,2297,3037,11927
> %N A046284 Concatenation of primes from 2 through p is a prime.
> ...
> %A A046284 Patrick De Geest (pdg(AT)worldofnumbers.com), Jun 15 1998.
> %E A046284 Additional comments from Robert G. Wilson v
> (rgwv(AT)rgwv.com), Sep 10 2001 (surely 5441 should be there?)
>
> OTOH the book may be wrong and the OEIS entries correct.  Can someone
> check?
>
> NJAS