A030997 -id:A030997 - cp's OEIS frontend

A068656 Duplicate of A030997.

2, 23, 5711, 2357, 711131719, 113127131137139149, 29313741434753

Offset: 1

dead

A030469 Primes which are concatenations of three consecutive primes.

5711, 111317, 171923, 313741, 414347, 8997101, 229233239, 239241251, 263269271, 307311313, 313317331, 317331337, 353359367, 359367373, 383389397, 389397401, 401409419, 409419421, 439443449, 449457461

Offset: 1

Patrick De Geest

a(n) = "p(k) p(k+1) p(k+2)" where p(k) is k-th prime

It is conjectured that sequence is infinite. - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009

			(1) 5=p(3), 7=p(4), 11=p(5) gives a(1).
(2) 7=p(4), 11=p(5), 13=p(6), but 71113 = 7 x 10159

Richard E. Crandall, Carl Pomerance: Prime Numbers, Springer 2005 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
John Derbyshire: Prime obsession, Joseph Henry Press, Washington, DC 2003 - from Eva-Maria Zschorn (e-m.zschorn(AT)zaschendorf.km3.de), Nov 09 2009
Marcus du Sautoy: Die Musik der Primzahlen. Auf den Spuren des groessten Raetsels der Mathematik, Beck, Muenchen 2004

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A030461, A167517, A132903, A068655, A030997, A030473, A086041, A099727.

Select[Table[FromDigits[Flatten[IntegerDigits/@{Prime[n],Prime[n+1],Prime[n+2]}]],{n,11000}],PrimeQ] (* Zak Seidov, Oct 16 2009 *)
concat[{a_,b_,c_}]:=FromDigits[Flatten[IntegerDigits/@{a,b,c}]]; Select[ concat/@ Partition[ Prime[ Range[200]],3,1],PrimeQ] (* Harvey P. Dale, Sep 06 2017 *)

for(i=1,999, isprime(p=eval(Str(prime(i),prime(i+1),prime(i+2)))) & print1(p," ")) \\ M. F. Hasler, Nov 10 2009

A132903 INTERSECT A000040. - R. J. Mathar, Nov 11 2009

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

A030996 Concatenation of n consecutive primes starting with the prime a(n) is a prime.

Original entry on oeis.org

2, 2, 5, 2, 7, 113, 29, 107, 211, 691, 23, 11, 5, 71, 277, 181, 13, 307, 43, 223, 43, 167, 31, 23, 151, 29, 67, 383, 19, 53, 53, 277, 541, 37, 103, 127, 947, 79, 4409, 1699, 13, 73, 31, 107, 751, 1187, 1069, 859, 67, 229, 293, 47, 1303, 2909, 1319

Offset: 1

Patrick De Geest, Warut Roonguthai

Paolo P. Lava, Table of n, a(n) for n = 1..250 (terms 1..100 from Zak Seidov)

Cf. A000040, A030997.

Edited by Charles R Greathouse IV, Apr 30 2010

A052077 Smallest prime formed by concatenating n consecutive increasing numbers, or 0 if no such prime exists.

Original entry on oeis.org

2, 23, 0, 4567, 1516171819, 0, 78910111213, 23456789, 0, 45678910111213, 129130131132133134135136137138139, 0, 567891011121314151617, 5051525354555657585960616263, 0, 128129130131132133134135136137138139140141142143

Offset: 1

Patrick De Geest, Jan 15 2000

Starting numbers in concatenations are given by A052079.

Paolo P. Lava, Table of n, a(n) for n = 1..43
C. Rivera, Prime Puzzle 78.

Cf. A052078, A052079, A052080, A030997.

f[n_] := If[Mod[n, 3] == 0, 0, Block[{k = 1}, While[d = FromDigits@ Flatten@ IntegerDigits[ Range[k, k + n - 1]]; !PrimeQ@ d, k++]; d]]; Array[f, 16] (* Robert G. Wilson v, Jun 29 2012 *)

Terms a(7)-a(15) are calculated by Carlos Rivera and Felice Russo

a(16) from Max Alekseyev, Jan 31 2010

A167518 Least reversible prime (A007500) which is a concatenation of n consecutive primes.

Original entry on oeis.org

2, 151157, 353359367, 139149151157, 101103107109113, 704517045770459704817048770489, 97101103107109113127, 1519315199152171522715233152411525915263, 382138233833384738513853386338773881, 9319932393379341934393499371937793919397

Offset: 1

M. F. Hasler, Nov 10 2009

Here the weaker definition of A007500 is used, but all terms > 2 known so far are also Emirps in the sense of A006567 (i.e. different from their reversal), so it is sufficient to change the first term to 13 in order to have a sequence of "true" emirps.

Is it possible to prove that all terms > 2 are in A006567?

Cf. A030997, A167517.

for(k=1,19,for(i=0,1e9, isprime( eval( p=concat( vector( k,j,Str( prime( i+j )))))) & isprime(eval(concat(vecextract(Vec(p),"-1..1")))) & break); print1(p,", "))

Edited by Charles R Greathouse IV, Apr 28 2010

a(9)-a(10) from Donovan Johnson, Sep 25 2011

A297935 Least prime k such that n concatenations of n+1 consecutive primes in base 2, starting from k, generate another prime in base 10.

Original entry on oeis.org

2, 2, 3, 2, 19, 53, 163, 53, 167, 31, 3, 37, 743, 97, 271, 17, 3, 41, 131, 691, 97, 181, 587, 523, 227, 211, 229, 3, 1697, 151, 1009, 23, 131, 151, 3137, 1621, 71, 439, 389, 521, 811, 1039, 179, 23, 311, 193, 227, 5869, 577, 6263, 31, 1901, 113, 1439, 1451, 107

Offset: 0

Paolo P. Lava, Jan 09 2018

			a(4) = 19 because the concatenation of 19, 23, 29, 31, 37 in base 2 is concat(concat(concat(concat(10011, 10111), 11101), 11111), 100101) that is the prime 41414629 in base 10 and 19 is the least prime to have this property.

Paolo P. Lava, Table of n, a(n) for n = 0..200

Cf. A000040, A030996, A030997.

with(numtheory): P:=proc(q) local a,b,c,i,k,n;
for n from 1 to q do for k from 1 to q do
a:=ithprime(k); b:=convert(a,binary,decimal);
for i from 1 to n-1 do a:=nextprime(a);
c:=convert(a,binary,decimal); b:=b*10^(ilog10(c)+1)+c; od;
a:=convert(b,decimal,binary); if isprime(a) then print(ithprime(k)); break; fi; od; od; end: P(10^3);

Table[Prime@ SelectFirst[Range[2^12], Function[k, PrimeQ@ FromDigits[Join @@ IntegerDigits[Prime@ Range[k, k + n], 2],2]]], {n, 0, 55}] (* Michael De Vlieger, Jan 09 2018 *)

eva(n) = subst(Pol(n), x, 10)
decimal(v, base) = my(w=[]); for(k=0, #v-1, w=concat(w, v[#v-k]*base^k)); sum(i=1, #w, w[i])
concat_primes(start, num) = my(v=[], s=""); forprime(p=start, , v=concat(v, [eva(binary(p))]); if(#v==num, break)); for(k=1, #v, s=concat(s, Str(v[k]))); eval(s)
a(n) = forprime(k=1, , if(ispseudoprime(decimal(digits(concat_primes(k, n+1)), 2)), return(k))) \\ Felix Fröhlich, Jan 09 2018

cp's OEIS Frontend

A068656 Duplicate of A030997.

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A030469 Primes which are concatenations of three consecutive primes.

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

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A030996 Concatenation of n consecutive primes starting with the prime a(n) is a prime.

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A052077 Smallest prime formed by concatenating n consecutive increasing numbers, or 0 if no such prime exists.

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A167518 Least reversible prime (A007500) which is a concatenation of n consecutive primes.

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PARI

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A297935 Least prime k such that n concatenations of n+1 consecutive primes in base 2, starting from k, generate another prime in base 10.

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