A046035 -id:A046035 - cp's OEIS frontend

A046034 Numbers whose digits are primes.

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 222, 223, 225, 227, 232, 233, 235, 237, 252, 253, 255, 257, 272, 273, 275, 277, 322, 323, 325, 327, 332, 333, 335, 337, 352, 353, 355, 357, 372, 373, 375, 377, 522, 523, 525, 527, 532

Offset: 1

Eric W. Weisstein

If n is represented as a zerofree base-4 number (see A084544) according to n=d(m)d(m-1)...d(3)d(2)d(1)d(0) then a(n) = Sum_{j=0..m} c(d(j))*10^j, where c(k)=2,3,5,7 for k=1..4. - Hieronymus Fischer, May 30 2012

According to A153025, it seems that 5, 235 and 72335 are the only terms whose square is also a term, i.e., which are also in the sequence A275971 of square roots of the terms which are squares, listed in A191486. - M. F. Hasler, Sep 16 2016

			a(100)   = 2277,
a(10^3)  = 55327,
a(9881)  = 3233232,
a(10^4)  = 3235757,
a(10922) = 3333333,
a(10^5)  = 227233257.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Robert Baillie and Thomas Schmelzer, Summing Kempner's Curious (Slowly-Convergent) Series, Mathematica Notebook kempnerSums.nb, Wolfram Library Archive, 2008.
Eric Weisstein's World of Mathematics, Smarandache Sequences.
Index entries for 10-automatic sequences.

Cf. A046035, A118950, A019546 (primes), A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A153025, A191486, A193238, A202267, A202268, A211681, A365471 (complement).

a046034 n = a046034_list !! (n-1)
a046034_list = filter (all (`elem` "2357") . show ) [0..]
-- Reinhard Zumkeller, Jul 19 2011

[n: n in [2..532] | Set(Intseq(n)) subset [2, 3, 5, 7]];  // Bruno Berselli, Jul 19 2011

Table[FromDigits /@ Tuples[{2, 3, 5, 7}, n], {n, 3}] // Flatten (* Michael De Vlieger, Sep 19 2016 *)

is_A046034(n)=Set(isprime(digits(n)))==[1] \\ M. F. Hasler, Oct 12 2013

def A046034(n):
    m = (3*n+1).bit_length()-1>>1
    return int(''.join(('2357'[(3*n+1-(1<<(m<<1)))//(3<<((m-1-j)<<1))&3] for j in range(m)))) # Chai Wah Wu, Feb 08 2023

A055642(a(n)) = A193238(a(n)). - Reinhard Zumkeller, Jul 19 2011
From Hieronymus Fischer, Apr 20, May 30 and Jun 25 2012: (Start)
a(n) = Sum_{j=0..m-1} ((2*b(j)+1) mod 8 + floor(b(j)/4) - floor((b(j)-1)/4))*10^j, where m = floor(log_4(3*n+1)), b(j) = floor((3*n+1-4^m)/(3*4^j)).
a(n) = Sum_{j=0..m-1} A010877(A005408(b(j)) + A002265(b(j)) - A002265(b(j)-1))*10^j.
Special values:
a(1*(4^n-1)/3) = 2*(10^n-1)/9.
a(2*(4^n-1)/3) = 1*(10^n-1)/3.
a(3*(4^n-1)/3) = 5*(10^n-1)/9.
a(4*(4^n-1)/3) = 7*(10^n-1)/9.
Inequalities:
a(n) <= 2*(10^log_4(3*n+1)-1)/9, equality holds for n = (4^k-1)/3, k>0.
a(n) <= 2*A084544(n), equality holds iff all digits of A084544(n) are 1.
a(n) > A084544(n).
Lower and upper limits:
lim inf a(n)/10^log_4(n) = (7/90)*10^log_4(3) = 0.48232167706987..., for n -> oo.
lim sup a(n)/10^log_4(n) = (2/9)*10^log_4(3) = 1.378061934485343..., for n -> oo.
where 10^log_4(n) = n^1.66096404744...
G.f.: g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(2 + z(j) + 2*z(j)^2 + 2*z(j)^3 - 7*z(j)^4)/(1-z(j)^4), where z(j) = x^4^j.
Also g(x) = (x^(1/3)*(1-x))^(-1) Sum_{j>=0} 10^j*z(j)^(4/3)*(1-z(j))*(2 + 3*z(j) + 5*z(j)^2 + 7*z(j)^3)/(1-z(j)^4), where z(j)=x^4^j.
Also: g(x) = (1/(1-x))*(2*h_(4,0)(x) + h_(4,1)(x) + 2*h_(4,2)(x) + 2*h_(4,3)(x) - 7*h_(4,4)(x)), where h_(4,k)(x) = Sum_{j>=0} 10^j*x^((4^(j+1)-1)/3)*x^(k*4^j)/(1-x^4^(j+1)). (End)
Sum_{n>=1} 1/a(n) = 1.857333779940977502574887651449435985318556794733869779170825138954093657197... (calculated using Baillie and Schmelzer's kempnerSums.nb, see Links). - Amiram Eldar, Feb 15 2024

More terms from Cino Hilliard, Aug 06 2006
Typo in second formula corrected by Hieronymus Fischer, May 12 2012
Two typos in example section corrected by Hieronymus Fischer, May 30 2012

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

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

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

A085557 Numbers that have more prime digits than nonprime digits.

Original entry on oeis.org

2, 3, 5, 7, 22, 23, 25, 27, 32, 33, 35, 37, 52, 53, 55, 57, 72, 73, 75, 77, 122, 123, 125, 127, 132, 133, 135, 137, 152, 153, 155, 157, 172, 173, 175, 177, 202, 203, 205, 207, 212, 213, 215, 217, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232

Offset: 1

Jason Earls, Jul 04 2003

Begins to differ from A046034 at the 21st term (which is the first 3-digit term).

			133 is in the sequence as the prime digits are 3 and 3 (those are two digits; counted with multiplicity) and one nonprime digit 1 and so there are more prime digits than nonprime digits. - _David A. Corneth_, Sep 06 2020

David A. Corneth, Table of n, a(n) for n = 1..10000

Cf. A193238, A046034, A046035, A118950, A019546, A203263, A035232, A039996, A085823, A052382, A084544, A084984, A017042, A001743, A001744, A014261, A014263, A202267, A202268, A211681.

is(n) = my(d = digits(n), c = 0); for(i = 1, #d, if(isprime(d[i]), c++)); c<<1 > #d \\ David A. Corneth, Sep 06 2020

from itertools import count, islice
def A085557_gen(startvalue=1): # generator of terms
    return filter(lambda n:len(s:=str(n))<(sum(1 for d in s if d in {'2','3','5','7'})<<1),count(max(startvalue,1)))
A085557_list = list(islice(A085557_gen(),20)) # Chai Wah Wu, Feb 08 2023

A099077 Numbers k such that pi(1).pi(2) ... pi(k-1).pi(k) is prime (dot between numbers means concatenation).

Original entry on oeis.org

5, 25, 2232, 4560

Offset: 1

Farideh Firoozbakht, Oct 23 2004

Number of digits of primes corresponding to the four known terms of this sequence are respectively 4,24,6127,13111. Next term is greater than 10250 and the prime corresponding to the next term has more than 32500 digits.

a(5) > 35000. - Michael S. Branicky, Nov 25 2024

			5 is in the sequence because pi(1).pi(2).pi(3).pi(4).pi(5)=1223 is prime.

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems connection.
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A000720, A046035, A099078, A099079, A099080.

A099078 Numbers k such that pi(k).pi(k-1) ... pi(3).pi(2) is prime (dot between numbers means concatenation).

Original entry on oeis.org

5, 22, 48, 317, 734, 5235, 12377

Offset: 1

Farideh Firoozbakht, Oct 23 2004

Number of digits of primes corresponding to the five known terms of this sequence are respectively 4, 21, 67, 605, 1633.

			5 is in the sequence because pi(5).pi(4).pi(3).pi(2) = 3221 is prime.

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems connection.
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A046035, A099077, A099079, A099080.

r:= 1: v:= 1: Res:= NULL:
for k from 3 to 6000 do
   if isprime(k) then r:= r+1 fi;
   v:= v + r*10^(1+ilog10(v));
   if isprime(v) then Res:= Res, k fi
od:
Res; # Robert Israel, Nov 20 2018

s = ""; Do[s = ToString[PrimePi[n]] <> s; k = ToExpression[s]; If[PrimeQ[k], Print[n]], {n, 2, 5235}] (* Ryan Propper, Aug 30 2005 *)

a(6) from Ryan Propper, Aug 30 2005

a(7) from Michael S. Branicky, Apr 29 2023

A099079 Numbers n such that phi(n).phi(n-1). ... .phi(2).phi(1) is prime (dots between numbers mean concatenation).

Original entry on oeis.org

2, 3, 9, 28, 30, 31, 51, 127, 208

Offset: 1

Farideh Firoozbakht, Oct 23 2004

Number of digits of primes corresponding to the nine known terms of this sequence are respectively 2,3,9,39,42,44,84,244,441.
If it exists, a(10) > 10362. - J.W.L. (Jan) Eerland, Aug 14 2022
If it exists, a(10) > 25000. - Michael S. Branicky, Aug 23 2024

			9 is in the sequence because phi(9).phi(8).phi(7).phi(6).phi(5).phi(4).phi(3).phi(2).phi(1) = 646242211 is prime.

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems connection.
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A046035, A099077, A099078, A099080.

Module[{nn=210,eph},eph=EulerPhi[Range[nn]];Position[Table[FromDigits[ Flatten[ IntegerDigits[Reverse[Take[eph,n]]]]],{n,nn}],?PrimeQ]]// Flatten (* _Harvey P. Dale, Apr 21 2020 *)
ParallelTable[If[PrimeQ[ToExpression[StringJoin[ToString[#]&/@Reverse[Table[EulerPhi[k],{k,1,n}]]]]],n,Nothing],{n,1,10^4}]//.{}->Nothing (* J.W.L. (Jan) Eerland, Aug 15 2022 *)

A099080 Numbers k such that sigma(k).sigma(k-1) ... sigma(2).sigma(1) is prime (dot between numbers means concatenation).

Original entry on oeis.org

2, 3, 66, 102

Offset: 1

Farideh Firoozbakht, Oct 23 2004

Numbers of digits of primes corresponding to the four known terms of this sequence are respectively 2, 3, 133, and 232.
A naive heuristic suggests that this sequence is infinite but extremely sparse. - Charles R Greathouse IV, Nov 05 2013
There are no more terms below 10000. - Charles R Greathouse IV, Nov 09 2013
There are no more terms below 20000. - Michael S. Branicky, Nov 25 2024

			3 is in the sequence because sigma(3).sigma(2).sigma(1) = 431 is prime.

Carlos Rivera, Puzzle 8. Primes by Listing, The Prime Puzzles & Problems connection.
Eric Weisstein's World of Mathematics, Integer Sequence Primes

Cf. A046035, A099077, A099078, A099079.

Module[{nn=110,d},d=DivisorSigma[1,Range[nn]];Select[Range[nn], PrimeQ[ FromDigits[ Flatten[IntegerDigits/@Reverse[Take[d,#]]]]]&]] (* Harvey P. Dale, Jul 25 2016 *)

s="1";for(n=2,1e3,s=Str(sigma(n),s);if(ispseudoprime(eval(s)), print1(n", "))) \\ Charles R Greathouse IV, Nov 05 2013

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 *)

cp's OEIS Frontend

A046034 Numbers whose digits are primes.

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

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

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A085557 Numbers that have more prime digits than nonprime digits.

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A099077 Numbers k such that pi(1).pi(2) ... pi(k-1).pi(k) is prime (dot between numbers means concatenation).

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A099078 Numbers k such that pi(k).pi(k-1) ... pi(3).pi(2) is prime (dot between numbers means concatenation).

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