A019519 -id:A019519 - cp's OEIS frontend

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

A072723 Integers which are exactly the concatenation of the first m odd numbers (A019519) divided by their sum (A000290 = m^2).

Original entry on oeis.org

1, 15, 16764334957, 3079163563531047898532266016633501

Offset: 1

Henry Bottomley, Jul 06 2002

A probability argument suggests that this sequence may be finite.

			a(1) = 1/1 = 1; a(2) = 135/(1+3+5) = 15; a(3) = 1357911131517/(1+3+5+7+9+11+13+15+17) = 16764334957; a(4) = 1357911131517192123252729313335373941/(1+3+5+7+9+11+13+15+17+19+21+23+25+27+29+31+33+35+37+39+41) = 3079163563531047898532266016633501.

Cf. A067122, A072724, A072725.

Select[Table[FromDigits[Flatten[IntegerDigits/@Range[1,2n+1,2]]]/Total[Range[1,2n+1,2]],{n,0,30}],IntegerQ] (* Harvey P. Dale, Mar 24 2023 *)

A067122 Floor[X/Y] where X = concatenation of first n odd numbers in increasing order (A019519) and Y = their sum (A000290 = n^2).

Original entry on oeis.org

1, 3, 15, 84, 543, 37719, 2771247, 212173614, 16764334957, 1357911131517, 112224060455966, 9429938413313834, 803497710956918416, 69281180179448577716, 6035160584520853881123, 530434035748903173145597

Offset: 1

Amarnath Murthy, Jan 08 2002

			a(4) = floor[1357/16] = floor[84.8125] =84.

Cf. A067112-A067121, A072723.

More terms from Henry Bottomley, Jul 07 2002

A260802 Odd numbers x = 2n - 1 such that the concatenation of A019519(n) and A038395(n-1) is prime.

Original entry on oeis.org

3, 13, 19, 21, 67

Offset: 1

Abhiram R Devesh, Jul 31 2015

			a(1) = 3 since 13_1 is prime;
a(2) = 13 since 135791113_1197531 is prime;
a(3) = 19 since 135791113151719_1715131197531 is prime.

Cf. A019519, A038395.

import sympy
n=1
while n>0:
    s=str(n)
    for m in range(n-2,0,-2):
        s=str(m)+s+str(m)
    p=int(s)
    if sympy.isprime(p)==True:
        print(n)
    n=n+2

A019520 a(n) is the concatenation of the first n positive even numbers.

Original entry on oeis.org

2, 24, 246, 2468, 246810, 24681012, 2468101214, 246810121416, 24681012141618, 2468101214161820, 246810121416182022, 24681012141618202224, 2468101214161820222426, 246810121416182022242628, 24681012141618202224262830, 2468101214161820222426283032

Offset: 1

R. Muller

H. Ibstedt, A Few Smarandache Sequences, Smarandache Notions Journal, Vol. 8, No. 1-2-3, 1997, 170-183.
F. Smarandache, "Collected Papers", Vol. II, Tempus Publ. Hse., Bucharest, Romania, 1996.
S. Smarandoiu, Convergence of Smarandache continued fractions, Abstract 96T-11-195, Abstracts Amer. Math. Soc., 17 (No. 4, 1996), 680.

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A019519 (similar, with odd numbers), A067095, A108728.

Table[FromDigits[Flatten[IntegerDigits/@(2Range[n])]],{n,20}] (* Harvey P. Dale, Mar 24 2013 *)

def a(n): return int("".join(str(2*i) for i in range(1, n+1)))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Dec 18 2021

Lim_{n->oo} A019519(n)/a(n) = 0 (see A067095). - Bernard Schott, Dec 18 2021

More terms from Erich Friedman

More terms from Harvey P. Dale, Mar 24 2013

A067095 a(n) = floor(X/Y) where X is the concatenation in increasing order of the first n even numbers and Y is that of the first n odd numbers.

Original entry on oeis.org

2, 1, 1, 1, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 18, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181, 181

Offset: 1

Amarnath Murthy, Jan 07 2002

For n > 1, the sequence is increasing and tends to infinity. Proof: for k>=1, when the last concatenated integer at the numerator A019520(n) has k digits, then a(n) > 10^(k-1) (see Krusemeyer reference). - Bernard Schott, Dec 06 2021

Values taken by this function are in A349960. - Bernard Schott, Dec 18 2021

			a(4) = floor(2468/1357) = floor(1.81871775976418570375829034635225) = 1.
a(20000) = 18175.

Mark I. Krusemeyer, George T. Gilbert, and Loren C. Larson, A Mathematical Orchard, Problems and Solutions, MAA, 2012, Problem 87, pp. 159-161.

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094.

Cf. A019519, A019520, A349960.

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2k]]; y = StringJoin[y, ToString[2k - 1]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 75} ]
With[{ev=Range[2,140,2],od=Range[1,139,2]},Table[Floor[FromDigits[ Flatten[ IntegerDigits/@ Take[ev,n]]]/FromDigits[Flatten[ IntegerDigits/@ Take[od,n]]]],{n,70}]] (* Harvey P. Dale, Aug 19 2011 *)

ae(n)=my(s=""); for(k=1, n, s=Str(s, 2*k)); eval(s); \\ A019520
ao(n)=my(s=""); for(k=1, n, s=Str(s, 2*k-1)); eval(s); \\ A019521
a(n) = ae(n)\ao(n); \\ Michel Marcus, Dec 07 2021

a(n) = floor(A019520(n)/A019519(n)).

More terms from Robert G. Wilson v, Jan 09 2002

A067096 Floor[X/Y] where X = concatenation in increasing order of first n even numbers and Y = that of first n natural numbers.

Original entry on oeis.org

2, 2, 2, 2, 19, 199, 1999, 19991, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916, 199916

Offset: 1

Amarnath Murthy, Jan 07 2002

Almost all terms appear only once. However, in the first 5000 terms, the term 2 appears 4 times in a row; the term 199916 appears 41 times in a row; the term 19991620000261183803815753482837892477715440187362570807 appears 401 times in a row; and a term with 556 digits (that begins with the same digits as the term that appears 401 times in a row) appears 4001 times in a row. Does this pattern continue? - Harvey P. Dale, Jul 04 2012

			a(10) = floor[ 2468101214161820/12345678910] = floor[199916.20000441271803658143252326] = 199916.

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094, A067095.

floor[A019520(n)/A019519(n)]

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2^k]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 40} ]
ccat[n_,i_]:=FromDigits[Flatten[IntegerDigits/@Range[i,n,i]]]; Table[ Floor[ ccat[2m,2]/ccat[m,1]],{m,40}] (* Harvey P. Dale, Jul 04 2012 *)

More terms from Robert G. Wilson v, Jan 09 2002

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

A067097 Floor[X/Y] where X = concatenation in increasing order of first n powers of 2 and Y = that of first n natural numbers.

Original entry on oeis.org

2, 2, 2, 20, 201, 2010, 201012, 20101226, 2010122457, 201012245610, 20101224560848, 2010122456084687, 201012245608468521, 201012245608468519453, 201012245608468519428723, 201012245608468519428463029, 2010122456084685194284602619644

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(6)= floor [ 248163264/123456] = floor[2010.13530326594090202177293] = 2010.

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094, A067095, A067096.

floor[A045507(n)/A019519(n)]

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2^k]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 20} ]
Table[Floor[FromDigits[Flatten[IntegerDigits/@(2^Range[n])]]/FromDigits[ Flatten[IntegerDigits/@Range[n]]]],{n,20}] (* Harvey P. Dale, Dec 30 2018 *)

More terms from Robert G. Wilson v, Jan 09 2002

More terms from Harvey P. Dale, Dec 30 2018

A038395 Concatenation of the first n odd numbers in reverse order.

Original entry on oeis.org

1, 31, 531, 7531, 97531, 1197531, 131197531, 15131197531, 1715131197531, 191715131197531, 21191715131197531, 2321191715131197531, 252321191715131197531, 27252321191715131197531, 2927252321191715131197531, 312927252321191715131197531

Offset: 1

M. I. Petrescu (mipetrescu(AT)yahoo.com)

a(n) starts with the digits of 2n-1. Indices of prime or probable prime terms are 1,2,37,62,409,...: see also A089922. - M. F. Hasler, Apr 13 2008

If n == 0 (mod 3), so is a(n). - Sergey Pavlov, Mar 29 2017

Mihaly Bencze [Beneze] and L. Tutescu, Some Notions and Questions in Number Theory, Sequence 3.

Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems, arXiv:math/0604019 [math.GM], 2006.

Cf. A089922, A109837, A038394-A038399, A019518, A019519.

Table[FromDigits[Flatten[IntegerDigits/@Join[Reverse[Range[1,n,2]]]]], {n,1,29,2}] (* Harvey P. Dale, Jun 02 2011 *)

t=""; for( n=1,10^3, ( t=eval( Str( 2*n-1,t))) & print(n" "t)) \\ M. F. Hasler, Apr 13 2008

def a(n): return int("".join(map(str, range(2*n-1, 0, -2))))
print([a(n) for n in range(1, 17)]) # Michael S. Branicky, Jan 31 2021

Edited and extended by M. F. Hasler, Apr 13 2008

Edited by T. D. Noe, Oct 30 2008

cp's OEIS Frontend

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

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A072723 Integers which are exactly the concatenation of the first m odd numbers (A019519) divided by their sum (A000290 = m^2).

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A067122 Floor[X/Y] where X = concatenation of first n odd numbers in increasing order (A019519) and Y = their sum (A000290 = n^2).

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A260802 Odd numbers x = 2n - 1 such that the concatenation of A019519(n) and A038395(n-1) is prime.

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A019520 a(n) is the concatenation of the first n positive even numbers.

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A067095 a(n) = floor(X/Y) where X is the concatenation in increasing order of the first n even numbers and Y is that of the first n odd numbers.

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A067096 Floor[X/Y] where X = concatenation in increasing order of first n even numbers and Y = that of first n natural numbers.

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Programs

Mathematica

Extensions

A048847 Primes formed by concatenation of first k odd numbers.

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