Smarandache-Wellin primes

Eric W. Weisstein eww at wolfram.com
Sun Jan 18 20:56:44 CET 2004

```On Sun, 18 Jan 2004, N. J. A. Sloane wrote:

> There is a conflict between the Crandall-Pomerance book and the OEIS.
>
> The book says the concatenation of the first 719 primes is a prime.

Then the book is wrong. What is should say is the the concatenation of the
first 128 primes (i.e., primes 2, 3, 5, ..., 719; primes *up to 719*) is
prime.  719 is the 128th prime.

ConsecutivePrimes[n_] := FromDigits[Flatten[IntegerDigits /@ Prime[Range[n]]]]
l={1,2,4,128,174,342,435,1429};

And@@PrimeQ/@ConsecutivePrimes/@l//Timing
{236.22 Second, True}

PrimeQ[ConsecutivePrimes[719]]//Timing
{0.01 Second,False}

A046284 is just Prime[A046035]

Prime[l]
{2,3,7,719,1033,2297,3037,11927}

Incidentally, you could add a comment to A046035 that there are no other
members <= 7837.

Cheers,
-E

> 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")
>
> %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
>

```