A069151 -id:A069151 - cp's OEIS frontend

A263959 Number of decimal digits in A069151(n).

1, 2, 4, 355, 499, 1171, 1543, 5719

Offset: 1

Eric W. Weisstein, Oct 30 2015

Subset of A227530 (Copeland-Erdős constant primes) corresponding to concatenation of a full (non-truncated) final prime.

a(9) > 459970. - Eric W. Weisstein (according to Mark Rodenkirch as of Nov 21 2015)

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

Cf. A019518, A046035, A069151.
Cf. A227529 (Copeland-Erdős constant primes).
Cf. A227530 (decimal digits in n-th Copeland-Erdős constant prime).

Cases[FromDigits /@ Rest[FoldList[Join, {}, IntegerDigits[Prime[Range[10^3]]]]], p_?PrimeQ :> IntegerLength[p]]

p=""; for(n=1, 1e4, p=concat(p, prime(n)); if(ispseudoprime( eval(p)), print1(#Str(p)", "))) \\ Altug Alkan, Oct 30 2015

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

Original entry on oeis.org

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)).

A046284 Primes p such that concatenation of primes from 2 through p is a prime.

Original entry on oeis.org

2, 3, 7, 719, 1033, 2297, 3037, 11927

Offset: 1

Patrick De Geest, Jun 15 1998

"w_n = (P_1)(P_2) ... (P_n) [A019518], by which notation we mean that w_n is constructed in decimal by simple concatenation of digits [much like the Almost Natural numbers (A007376)]. For example, the first few w_n are 2, 23, 235, 2357, 235711, ... ." - Crandall and Pomerance

			7 is a member, since 2357 is a prime.

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective, Springer, NY, 2001; see p. 72. [The 2002 printing states incorrectly that 5441 is a term.]

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 Number

Cf. A019518, A033308, A069151. a(n) = prime(A046035(n)).

a = ""; Do[a = StringJoin[a, ToString[ Prime[n]]]; If[ PrimeQ[ ToExpression[a]], Print[n]], {n, 1, 1429}]

Additional comments from Robert G. Wilson v, Sep 10 2001

A227529 Copeland-Erdős constant primes (concatenation of entries (digits) of A033308 is prime).

Original entry on oeis.org

2, 23, 2357, 23571113171

Offset: 1

Eric W. Weisstein, Jul 14 2013

Primes obtained by taking consecutive decimal digits (starting with the first) of the Copeland-Erdős constant.
The next terms are too big to display: a(5) = 235711131...6917017097 (353 digits), a(6) = 235711131...1701709719 (355 digits), ...
See A227530 for an equivalent but more compact way of listing the terms, namely, by giving the number of digits of the constant A033308 that must be taken to get a prime. - M. F. Hasler, Apr 24 2017

Eric W. Weisstein, Table of n, a(n) for n = 1..7
Hans Havermann and Eric W. Weisstein (first 11 terms), Table of n, a(n) for n = 1..14
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant

Cf. A227530 (number of decimal digits in the n-th Copeland-Erdős prime).
Cf. A019518, A069151.
Cf. A033308 (Decimal expansion of Copeland-Erdős constant: concatenate primes).

A240563 Primes formed from concatenation of higher primes onto the previous entry until prime, starting from 2.

Original entry on oeis.org

2, 23, 2311, 231131, 23113147, 23113147229, 23113147229251, 23113147229251577, 23113147229251577857, 23113147229251577857859, 23113147229251577857859911, 231131472292515778578599111123, 2311314722925157785785991111231223

Offset: 1

Bill McEachen, Apr 07 2014

This generates a monotonically increasing sequence, nicely spread out, likely infinite. By altering the starting prime value, a family of such sequences can easily be generated.
Derived from A080155. - T. D. Noe, Apr 11 2014
From the first 155 points, with x = #digits, y = sequence pointer y~ A*x^B with (A, B) = (0.6624, 0.8106). This indicates a 100-digit prime in the vicinity of y = 28 for example. - Bill McEachen, Apr 13 2014
Only from the first 100 entries, it would appear that an upper bound on the number of digits in a(n) is A092777(n). - Bill McEachen, Sep 15 2015

			Begin from 2.
Next we try 23 - it is prime, this sets next iteration (23 is the "constant" part), upon which we try higher primes.
Next we try 235 - composite; next we try 237 - composite; next we try 2311 - prime, this sets next iteration (2311 now becomes the "constant" part), upon which we try higher primes.
Next we try 231113 - composite; next we try 231117 - composite; ...; next we try 231131 - prime, this sets next iteration (231131 now becomes the "constant" part), upon which we try higher primes.
Next we try 23113147 - prime, this sets next iteration (23113147 now becomes the "constant" part), upon which we try higher primes.

Bill McEachen, Table of n, a(n) for n = 1..100

Cf. A069151 (variant).

Cf. A080155 (primes used in concatenation).

X:= 2: p:= 3: a[1]:= 2:
for i from 2 to 30 do
  while not isprime(X*10^(1+ilog10(p))+p) do
     p:= nextprime(p)
  od:
  X:= X*10^(1+ilog10(p))+p;
  a[i]:= X;
  p:= nextprime(p);
od:
seq(a[i],i=1..30); # Robert Israel, Sep 15 2015

s[1] = 2; s[n_] := s[n] = Block[{d = Flatten[IntegerDigits /@ Array[s, n-1]], p = NextPrime@s[n - 1]}, While[! PrimeQ@ FromDigits@ Join[d, IntegerDigits@p], p = NextPrime@p]; p]; a[n_] := FromDigits@ Flatten[ IntegerDigits /@ Array[s, n]]; Array[a, 10] (* Giovanni Resta, Apr 09 2014 *)

print1(N=2); p=3; for(n=2,10, while(!isprime(eval(Str(N,p))), p=nextprime(p+1)); N=eval(Str(N,p)); p=nextprime(p+1); print1(", "N)) \\ Charles R Greathouse IV, Apr 09 2014

a(7)-a(13) from Giovanni Resta, Apr 09 2014

A227530 Integer lengths of the n-th Copeland-Erdős prime (concatenation of the first n entries (digits) of A033308 is prime).

Original entry on oeis.org

1, 2, 4, 11, 353, 355, 499, 1171, 1543, 5719, 11048, 68433, 97855, 292447

Offset: 1

Eric W. Weisstein, Jul 14 2013

a(15) > 300000. - Eric W. Weisstein (according to Mark Rodenkirch as of Dec 13 2015)
The larger of these are probable primes.
a(15) > 5*10^5 according to Mark Rodenkirch, June 2016. - M. F. Hasler, Apr 24 2017

Mark Rodenkirch (a.k.a. "rogue"), Copeland-Erdos Constant Primes, MersenneForum.org, June 2016.
Eric Weisstein's World of Mathematics, Copeland-Erdős Constant Digits

Cf. A227529 (primes corresponding to a(n) digits of the Copeland-Erdős constant), A019518, A069151.

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

a(12) = 68433 from Eric W. Weisstein, Aug 17 2013
a(13) = 97855 from Eric W. Weisstein, Oct 24 2015
a(14) = 292447 from Eric W. Weisstein, Dec 17 2015

A276481 Numbers k such that b(k) is prime, where b(1) = prime(1) = 2, b(n) = 10*b(n-1) + (prime(n) mod 10).

Original entry on oeis.org

1, 2, 4, 13, 16, 17, 28, 34, 90, 100, 132, 331, 534, 7923, 10157, 40197

Offset: 1

Thomas Ordowski, Sep 05 2016

Primes in the sequence b(n) are 2, 23, 2357, 2357137939171, ...

a(17) > 50000. - Michael S. Branicky, Aug 12 2025

Cf. A007652, A069151, A386964.

Res:= NULL: p:= 0: b:= 0:
for n from 1 to 600 do
  p:= nextprime(p);
  b:= 10*b + (p mod 10);
  if isprime(b) then Res:= Res, n fi
od:
Res; # Robert Israel, Sep 05 2016

b[1] = Prime@ 1; b[n_] := b[n] = 10 b[n - 1] + Mod[Prime@ n, 10]; Select[Range[10^3], PrimeQ@ b[#] &] (* Michael De Vlieger, Sep 06 2016 *)

b(n) = if (n==1, 2, 10*b(n-1) + (prime(n) % 10));
isok(n) = isprime(b(n)); \\ Michel Marcus, Sep 05 2016

list(lim)=my(v=List(),s,n); forprime(p=2,, if(n++>lim, return(Vec(v))); if(ispseudoprime(s=10*s+p%10), listput(v, n))) \\ Charles R Greathouse IV, Sep 05 2016

{k | A386964(k) is prime}. - Michael S. Branicky, Aug 12 2025

a(7)-a(13) from Michel Marcus, Sep 05 2016
a(14) from Robert Israel, Sep 05 2016
a(15)-a(16) from Michael S. Branicky, Aug 11 2025

A283802 Numbers k such that the concatenation of the first k odd composite numbers is a prime.

Original entry on oeis.org

21, 73, 132

Offset: 1

XU Pingya, Mar 17 2017

Indices k for which A283801(k) is prime.
A283801(21) = 91521252733353945495155576365697577818587 is a 41-digit prime;
A283801(73) = 91521...247249253 is a 193-digit prime;
A283801(132) = 91521...423425427 is a 370-digit prime.
Next term, if it exists, will be more than 5028.
a(4) > 25000, if it exists. - Michael S. Branicky, Apr 30 2025

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A007908, A019518, A046035, A069151, A071904, A283801.

k = 2; cc = oc = 0; lst = {}; While[k < 428, If[OddQ@k && !PrimeQ@k, cc = cc*10^IntegerLength@k + k; oc++; If[PrimeQ[cc], AppendTo[lst, oc]]]; k++]; lst  (* Robert G. Wilson v, Mar 18 2017 *)
Module[{nn=501,ocm},ocm=Select[Range[9,nn,2],CompositeQ];Select[ Range[ Length[ ocm]],PrimeQ[FromDigits[Flatten[IntegerDigits/@Take[ocm,#]]]]&]] (* Harvey P. Dale, Sep 02 2022 *)

A379426 Prime terms in A287353.

Original entry on oeis.org

2, 23, 2357, 23581, 2358247, 235824913, 235824916247, 2358249162515829584909, 235824916251582958491829824917162558516292249258249589629182571583855789, 2358249162515829584918298249171625585162922492582495896291825715838558298516316558918298250261

Offset: 1

Ya-Ping Lu, Dec 22 2024

nonn

Primes Sum_{i=1..k} 10^(k-i)*prime(i) for some k.

			k    prime(k)  A287353(k)  n    a(n)
---  --------  ----------  ---  -------
1    2         2           1    2
2    3         23          2    23
3    5         235
4    7         2357        3    2357
5    11        23581       4    23581
6    13        235823
7    17        2358247     5    2358247

Cf. A013918, A069151, A287353.

from sympy import isprime, nextprime
m = p = 0
while p < 500:
    p = nextprime(p); m = 10*m + p
    if isprime(m): print(m, end = ', ')

cp's OEIS Frontend

A263959 Number of decimal digits in A069151(n).

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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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A046284 Primes p such that concatenation of primes from 2 through p is a prime.

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A227529 Copeland-Erdős constant primes (concatenation of entries (digits) of A033308 is prime).

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A240563 Primes formed from concatenation of higher primes onto the previous entry until prime, starting from 2.

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Maple

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A227530 Integer lengths of the n-th Copeland-Erdős prime (concatenation of the first n entries (digits) of A033308 is prime).

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A276481 Numbers k such that b(k) is prime, where b(1) = prime(1) = 2, b(n) = 10*b(n-1) + (prime(n) mod 10).