A046036 -id:A046036 - 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

A048847 Primes formed by concatenation of first k odd numbers.

Original entry on oeis.org

13, 135791113151719, 135791113151719212325272931, 135791113151719212325272931333537394143454749515355575961636567

Offset: 1

N. J. A. Sloane, Charles T. Le (charlestle(AT)yahoo.com)

The next term (a(5)) has 93 digits. - Harvey P. Dale, Mar 05 2013

a(6) has 9725 digits (see A066811(6) or A046036(6)). - Michel Marcus, Jan 31 2014

R. W. Stephan, Factors and Primes in Two Smarandache Sequences, Smarandache Notions Journal, second edition, Vol. 9, No. 1-2, 1998, 5-11.

M. Fleuren, Smarandache Concatenated Odds.
Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A019519, A046036.

Select[Table[FromDigits[Flatten[IntegerDigits/@Range[1,2n+1,2]]],{n,40}], PrimeQ] (* Harvey P. Dale, Mar 05 2013 *)

Edited by Charles R Greathouse IV, Apr 23 2010

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)).

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A019519 Concatenate odd numbers.

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A048847 Primes formed by concatenation of first k odd numbers.

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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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Programs

Mathematica

Python

Formula

Extensions

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

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Author

Keywords

Links

Crossrefs

Programs

PARI

Formula