A048847 -id:A048847 - cp's OEIS frontend

A019519 Concatenate odd numbers.

1, 13, 135, 1357, 13579, 1357911, 135791113, 13579111315, 1357911131517, 135791113151719, 13579111315171921, 1357911131517192123, 135791113151719212325, 13579111315171921232527, 1357911131517192123252729, 135791113151719212325272931

Offset: 1

R. Muller

S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

X. Chen and M. Le, The Module Periodicity of Smarandache Concatenated Odd Sequence, Smarandache Notions Journal, Vol. 9, No. 1-2. 1998, 103-104.
H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
F. Smarandache, Definitions, Solved and Unsolved Problems, Conjectures, ... , edited by M. Perez, Xiquan Publishing House 2000.
F. Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse., Bucharest, Romania, 1996.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Index entries for sequences related to Most Wanted Primes video

Primes are in A048847, while their indices are in A046036.

Cf. A019520 (similar, with even numbers), A067095.

a:= proc(n) a(n):= `if`(n=1, 1, parse(cat(a(n-1), 2*n-1))) end:
seq(a(n), n=1..20); # Alois P. Heinz, Jan 13 2021

nn=20;With[{odds=Range[1,2nn+1,2]},Table[FromDigits[Flatten[ IntegerDigits/@ Take[odds,n]]],{n,nn}]] (* Harvey P. Dale, Aug 14 2014 *)

a(n)=my(s=""); for(k=1, n, s=Str(s, 2*k-1)); eval(s); \\ Michel Marcus, Dec 07 2021

def a(n): return int("".join(map(str, range(1, 2*n, 2))))
print([a(n) for n in range(1, 18)]) # Michael S. Branicky, Jan 13 2021

Sequence grows like 10^K, where K = 2 + floor(log_10(n)) + floor(log_10(a(n-1))). More generally we may consider a(n)= F(a(n-1),n)*B^K + G(a(n-1),n); K = floor(log_B H(a(n-1),n)); F(a(n-1),n); G(a(n-1),n); H(a(n-1),n) integer polynomials; B integer. - Ctibor O. Zizka, Mar 08 2008

Lim_{n->oo} a(n)/A019520(n) = 0 (see A067095). - Bernard Schott, Dec 07 2021

More terms from Erich Friedman

More terms from Harvey P. Dale, Aug 14 2014

A046036 Indices of the concatenation of the first k odd numbers (A019519) which are primes.

Original entry on oeis.org

2, 10, 16, 34, 49, 2570

Offset: 1

Eric W. Weisstein

No others with k <= 12000. - Eric W. Weisstein, Mar 01 2004

No others with k <= 25000. - Michael S. Branicky, Aug 28 2025

H. Ibstedt, A Few Smarandache Integer Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, pp. 171-183.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences
Eric Weisstein's World of Mathematics, Integer Sequence Primes
Index entries for sequences related to Most Wanted Primes video

Cf. A019519, A048847, A066811.

Module[{nn=50,odds},odds=Range[1,2nn+1,2];Position[Table[ FromDigits[ Flatten[IntegerDigits/@Take[odds,n]]],{n,nn}],?(PrimeQ[#]&)]]// Flatten (* The program generates the first 5 terms of the sequence. To generate the 6th, increase the value of nn to 2570 or more, but the program will take a long time to run. *) (* _Harvey P. Dale, Jul 21 2021 *)

a(n) = (A066811(n)+1)/2. - Michel Marcus, Jan 31 2014

A066811 Numbers k such that the concatenation of odd numbers from 1 to k is a prime.

Original entry on oeis.org

3, 19, 31, 67, 97, 5139

Offset: 1

Patrick De Geest, Jan 20 2002

a(7) > 50000. - Michael S. Branicky, Aug 28 2025

			19 is a term because 135791113151719 is a prime.

Cf. A000040, A005408, A046036, A048847.

p = ""; Do[p = p <> ToString[2*n+1]; If[PrimeQ[ToExpression[p]], Print[2*n+1]], {n, 0, 2569}] (* Ryan Propper, Aug 26 2005 *)

from sympy import isprime
def agen():
  k, str1tok = 1, '1'
  while True:
    if isprime(int(str1tok)): yield k
    k, str1tok = k + 2, str1tok + str(k + 2)
g = agen()
print([next(g) for i in range(5)]) # Michael S. Branicky, Mar 19 2021

a(n) = 2*A046036(n) - 1. - Michel Marcus, Jan 31 2014

a(6) from Ryan Propper, Aug 26 2005

A138965 Least prime factor of concatenation of first n odd numbers.

Original entry on oeis.org

1, 13, 3, 23, 37, 3, 11617, 5, 3, 135791113151719, 29, 3, 5, 11, 3, 135791113151719212325272931, 17, 3, 7, 13, 3, 131, 5, 3, 11, 25471443030907588399109, 3, 5, 7, 3, 181, 41, 3, 135791113151719212325272931333537394143454749515355575961636567, 19, 3, 40351, 5, 3, 7, 11, 3, 5, 57041, 3, 351269, 11, 3, 135791113151719212325272931333537394143454749515355575961636567697173757779818385878991939597

Offset: 1

M. F. Hasler, Apr 14 2008

Sean A. Irvine, Table of n, a(n) for n = 1..300
F. Smarandache, Sequences of numbers involved in unsolved problems, arXiv:math/0604019

Cf. A019519, A046036, A048847; A109837, A104644.

t=1; for( n=2,99, print1( factor( eval( t=Str( t,2*n-1 )))[1,1], ", "))

A138965(n) = A020639(A019519(n)) (= 3 if n = 0 (mod 3)).

A350153 Prime numbers created by concatenating all numbers 1 through k for some k > 1, then continuing to concatenate all numbers from k-1 towards 1. Primes are added to the sequence as they are found as k increases.

Original entry on oeis.org

12343, 1234543, 12345678910987, 12345678910987654321, 12345678910111213141516171819202122212019181716151413, 12345678910111213141516171819202122232425262728293029

Offset: 1

Patrick Quam, Dec 16 2021

A173426(n) is the concatenation of all numbers from 1 up to k and then back down to 1. The prime terms of A173426 have been called "memorable primes" (see the Numberphile video). These "unmemorable primes" are a superset created by concatenating 1..k in ascending order followed by concatenating the numbers k-1..1 in descending order. Any primes found during either concatenation process are added to the sequence (e.g., k = 5, 1234543 is included. If 12345 were prime, it would be included as well).

			For k=10, the first prime obtained by concatenating the numbers 1..10 and then concatenating the first one or more numbers from 9..1 is 12345678910987.

Patrick Quam, Table of n, a(n) for n = 1..29
Brady Haran and N. J. A. Sloane, The Most Wanted Prime Number, Numberphile video (2021).

Cf. A048847, A007908, A173426, A260343, A323532.

select(isprime, [seq(seq(parse(cat($1..n, n-i$i=1..t)),
                t=0..n-1), n=1..30)])[];  # Alois P. Heinz, Dec 19 2021

lst={};Table[s=Flatten[IntegerDigits/@Range@n];k=n-1;
While[k!=-1,If[PrimeQ[p=FromDigits@s],AppendTo[lst,p]];s=Join[s,IntegerDigits@k];k--],{n,100}];lst (* Giorgos Kalogeropoulos, Dec 17 2021 *)

from itertools import count, chain, islice, accumulate
from sympy import isprime
def A350153gen(): return filter(lambda p:isprime(p),(int(s) for n in count(1) for s in accumulate(str(d) for d in chain(range(1,n+1),range(n-1,0,-1)))))
A350153_list = list(islice(A350153gen(),20)) # Chai Wah Wu, Dec 20 2021

A104239 Number of distinct prime factors of 135...(2n-1) (concatenation of n odd numbers).

Original entry on oeis.org

0, 1, 2, 2, 2, 3, 2, 5, 3, 1, 4, 3, 3, 2, 3, 1, 5, 5, 7, 4, 5, 5, 4, 5, 5, 2, 4, 5, 4, 6, 5, 3, 7, 1, 4, 4, 3, 4, 3, 6, 5, 7, 5, 7, 8, 4, 7, 7, 1, 6

Offset: 1

Parthasarathy Nambi, Apr 16 2005

Interestingly, 135791113151719 is prime.

			The number of distinct prime factors of 13 is 1 (a prime) - the second term in the sequence.
The number of distinct prime factors of 135 is 2 - the third term in the sequence.
The number of distinct prime factors of 1357 is 2 - the fourth term in the sequence.

Eric Weisstein's World of Mathematics, Consecutive Number sequences

Cf. A001221, A048847.

Join[{0},Table[PrimeNu[FromDigits[Flatten[IntegerDigits/@Range[1,2n+1,2]]]],{n,50}]] (* Harvey P. Dale, Jun 15 2019 *)

Corrected and extended by Franklin T. Adams-Watters, Sep 01 2006

Edited by N. J. A. Sloane, May 21 2008 at the suggestion of R. J. Mathar

cp's OEIS Frontend

A019519 Concatenate odd numbers.

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A046036 Indices of the concatenation of the first k odd numbers (A019519) which are primes.

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A066811 Numbers k such that the concatenation of odd numbers from 1 to k is a prime.

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A138965 Least prime factor of concatenation of first n odd numbers.

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A350153 Prime numbers created by concatenating all numbers 1 through k for some k > 1, then continuing to concatenate all numbers from k-1 towards 1. Primes are added to the sequence as they are found as k increases.

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A104239 Number of distinct prime factors of 135...(2n-1) (concatenation of n odd numbers).

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Mathematica

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