A046284 -id:A046284 - cp's OEIS frontend

A019518 Smarandache-Wellin numbers: a(n) is the concatenation of first n primes (written in base 10).

2, 23, 235, 2357, 235711, 23571113, 2357111317, 235711131719, 23571113171923, 2357111317192329, 235711131719232931, 23571113171923293137, 2357111317192329313741, 235711131719232931374143, 23571113171923293137414347

Offset: 1

R. Muller

			E.g. a(6) = 2_3_5_7_11_13 = 23571113.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 72. [The 2002 printing states incorrectly that a(719) is prime. Cf. A046035.] This book uses the name "Smarandache-Wellin numbers", referring to a 1998 private communication from P. Wellin.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
M. Le, On Smarandache Concatenated Sequences I: Prime Power Sequences, Smarandache Notions Journal, Vol. 9, No. 1-2, 1998, 129-130.
S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

Reinhard Zumkeller, Table of n, a(n) for n = 1..300
M. Fleuren, Factoring of the Smarandache Concatenated Prime Sequence.
F. Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse., Bucharest, Romania, 1996.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant
Index entries for sequences related to Most Wanted Primes video

For the primes in this sequence see A069151. For where the primes occur see A046035.

Cf. A000040, A038394, A046284, A068670 (number of digits).

a019518 n = a019518_list !! (n-1)
a019518_list = map read $ scanl1 (++) $ map show a000040_list :: [Integer]
-- Reinhard Zumkeller, Mar 03 2014

[Seqint(Reverse(&cat[Reverse(Intseq(NthPrime(k))): k in [1..n]])): n in [1..20]]; // Vincenzo Librandi, Aug 23 2015

ConsecutivePrimes[n_] := FromDigits[Flatten[IntegerDigits /@ Prime[Range[n]]]] (* Eric W. Weisstein *)
Table[FromDigits[Flatten[IntegerDigits[Prime[Range[i]]]]],{i,15}] (* Jayanta Basu, May 30 2013 *)

s="";for(n=1,30,print1(s=Str(s,prime(n))",")) \\ Cino Hilliard; simplified by M. F. Hasler, Oct 06 2013

A019518(n)=eval(concat(concat([""],primes(n)))) \\ Faster than concat(apply(s->Str(s),primes(n))) or forprime(...s=Str(s,p)). - M. F. Hasler, Oct 06 2013

Definition edited by N. J. A. Sloane, Jul 02 2017

A046035 Numbers k such that the concatenation of the first k primes (A019518) is a prime.

Original entry on oeis.org

1, 2, 4, 128, 174, 342, 435, 1429

Offset: 1

Eric W. Weisstein

No other terms with k <= 34736. - Eric W. Weisstein, Oct 30 2015

			4 is a term since 2357 is a prime. [Corrected by Ed Murphy (emurphy42(AT)socal.rr.com), May 15 2007]

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 72. [The 2002 printing states incorrectly that 719 is a term.]

M. Fleuren, Smarandache Concatenated Primes.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Eric Weisstein's World of Mathematics, Smarandache-Wellin Prime
Index entries for sequences related to Most Wanted Primes video

Cf. A019518, A069151, A046284.

Cf. A033308 (Decimal expansion of Copeland-Erdős constant: concatenate primes).

max = 1500; With[{primes = Prime[Range[max]]}, Flatten[Position[ Table[ FromDigits[Flatten[IntegerDigits/@Take[primes, n]]], {n, max}], ?PrimeQ]]] (* _Harvey P. Dale, Dec 17 2013 *)
Position[FromDigits /@ Rest[FoldList[Join, {}, IntegerDigits[Prime[Range[ 10^3]]]]], ?PrimeQ] // Flatten (* _Eric W. Weisstein, Oct 30 2015 *)

p=""; for(n=1, 2000, p=concat(p, prime(n)); if(ispseudoprime(eval(p)), print1(n", "))) \\ Altug Alkan, Oct 30 2015

a(n) = A000720(A046284(n)), or A046284(n) = prime(a(n)).

A069151 Concatenations of consecutive primes, starting with 2, that are also prime.

Original entry on oeis.org

2, 23, 2357

Offset: 1

Joseph L. Pe, Apr 08 2002

Primes in A019518.
The next term is the 355-digit number 2357111317192329313741434753...677683691701709719 which is too large to include here. See A046035, A046284.
The term after the 355-digit term has 499 digits, and the next two terms after that have 1171 and 1543 digits respectively. - Harvey P. Dale, Oct 03 2024

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, 2nd ed., Springer, NY, 2005; see p. 78. [The 2002 printing states incorrectly that 2357...5441 is prime.]

Jens Kruse Andersen, Table of n, a(n) for n = 1..5
M. Fleuren, Smarandache Concatenated Primes
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Eric Weisstein's World of Mathematics, Smarandache-Wellin Number
Eric Weisstein's World of Mathematics, Smarandache-Wellin Prime
Wikipedia, Smarandache-Wellin number
Index entries for sequences related to Most Wanted Primes video

Cf. A019518.
Cf. A046035 (Numbers n such that the concatenation of the first n primes is prime)
Cf. A046284 (Primes p such that concatenation of primes from 2 through p is a prime).
Cf. A030997 (Smallest prime which is a concatenation of n consecutive primes).

Cases[FromDigits /@ Rest[FoldList[Join, {}, IntegerDigits[Prime[ Range[10^3]]]]], ?PrimeQ] (* _Eric W. Weisstein, Oct 30 2015 *)
Select[Table[FromDigits[Flatten[IntegerDigits/@Prime[Range[n]]]],{n,500}],PrimeQ] (* Harvey P. Dale, Oct 03 2024 *)

s=""; for(n=1, 200, s=concat(s, prime(n)); if(ispseudoprime( eval(s)), print1(s", "))) \\ Jens Kruse Andersen, Jun 26 2014

from sympy import isprime, nextprime
def afind(terms, verbose=False):
  n, p, pstr = 0, 2, "2"
  while n < terms:
    if isprime(int(pstr)): n += 1; print(n, int(pstr))
    p = nextprime(p); pstr += str(p)
afind(5) # Michael S. Branicky, Feb 23 2021

Edited by Robert G. Wilson v, Apr 11 2002

Entry revised Jan 18 2004

A100003 Prime numbers p such that the concatenation of all odd primes up through p in decreasing order is prime.

Original entry on oeis.org

3, 5, 17, 89, 383, 8831

Offset: 1

Farideh Firoozbakht, Nov 06 2004

Next term is greater than 4400th prime and the prime corresponding to the next term has more than 20000 digits. Number of digits of primes corresponding to the six known terms of the sequence are respectively 1, 2, 9, 43, 198, 4202.
We can see the prime corresponding to 383 (the 5th term of the sequence) in the page related to puzzle 8 of the website of Carlos Rivera.
a(7) > prime(28800) = 335033. - Giovanni Resta, Apr 01 2013

			17 is in the sequence because 17.13.11.7.5.3 is prime (dot between numbers means concatenation).

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles and Problems Connection.

Cf. A046284, A099070, A099071, A099073.

The actual prime concatenations in A092448 and the original concatenations in A092447. - Dmitry Kamenetsky, Mar 02 2009

Do[If[PrimeQ[(v={};Do[v=Join[v, IntegerDigits[Prime[n-j+1]]], {j, n-1}];FromDigits[v])], Print[Prime[n]]], {n, 2, 4413}]
Prime[#]&/@Select[Range[100],PrimeQ[FromDigits[Flatten[IntegerDigits/@ Prime[Range[#,2,-1]]]]]&] (* To generate a(6) increase the Range by 1000, but the program will run a long time. *) (* Harvey P. Dale, Nov 27 2015 *)

A099070 Numbers k such that the concatenation of all nonprime natural numbers up to k with decreasing order is prime.

Original entry on oeis.org

4, 5, 6, 7, 8, 9, 26, 1752, 1753

Offset: 1

Farideh Firoozbakht, Nov 04 2004

If k is in the sequence and k+1 is prime then k+1 is also in the sequence. Next term is greater than 5450 and the prime corresponding to the next term has more than 18000 digits. Number of digits of primes corresponding to the nine known terms of the sequence are respectively 2,2,3,3,4,5,29,5010 & 5010.

a(10) > 20000. - Michael S. Branicky, Nov 25 2024

			9 is in the sequence because all nonprime natural numbers up to 9 are 1,4,6,8 & 9 and 98641 is prime.

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems connection.

Cf. A046284, A099071.

Do[If[PrimeQ[(v={};Do[If[ !PrimeQ[n+1-j], v=Join[v, IntegerDigits [n+1-j]]], {j, n}];FromDigits[v])], Print[n]], {n, 5450}]

A099071 Composite numbers k such that the concatenation of all nonprime positive integers up to k in decreasing order is prime.

Original entry on oeis.org

4, 6, 8, 9, 26, 1752

Offset: 1

Farideh Firoozbakht, Nov 06 2004

The terms of this sequence are composite terms of the sequence A099070 with the same order. Next term is greater than 6000 and the prime corresponding to the next term has more than 20000 digits. Number of digits of primes corresponding to the six known terms of the sequence are respectively 2, 3, 4, 5, 29, and 5010.

			26 is a term: 26 is composite; nonprimes up to 26 are 1, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26; and 26252422212018161514121098641 is prime.

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles and Problems Connection.

Cf. A099070, A100003, A046284.

Do[If[ !PrimeQ[n]&&PrimeQ[(v={};Do[If[ !PrimeQ[n+1-j], v=Join[v, IntegerDigits[n+1-j]]], {j, n}];FromDigits[v])], Print[n]], {n, 6013}]
cnpQ[n_]:=PrimeQ[FromDigits[Flatten[IntegerDigits/@Select[Range[n,1,-1],!PrimeQ[#]&]]]]; Select[Range[1800],!PrimeQ[#]&&cnpQ[#]&] (* Harvey P. Dale, Jul 19 2020 *)

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A019518 Smarandache-Wellin numbers: a(n) is the concatenation of first n primes (written in base 10).

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A046035 Numbers k such that the concatenation of the first k primes (A019518) is a prime.

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A069151 Concatenations of consecutive primes, starting with 2, that are also prime.

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A100003 Prime numbers p such that the concatenation of all odd primes up through p in decreasing order is prime.

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A099070 Numbers k such that the concatenation of all nonprime natural numbers up to k with decreasing order is prime.

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A099071 Composite numbers k such that the concatenation of all nonprime positive integers up to k in decreasing order is prime.

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