A019520 -id:A019520 - cp's OEIS frontend

A072724 Integers which are exactly the concatenation of the first m even numbers (A019520) divided by their sum (A002378 = m^2+m).

1, 4, 8227, 3427918353

Offset: 1

Henry Bottomley, Jul 06 2002

A probability argument suggests that this sequence may be finite.

			a(1) = 2/2 =1; a(2) = 24/(2+4) = 4; a(3) = 246810/(2+4+6+8+10) = 8227; a(4) = 246810121416/(2+4+6+8+10+12+14+16).

Cf. A067121, A072723, A072725.

With[{eds=Range[2,1500,2]},Select[Table[FromDigits[Flatten[ IntegerDigits/@ Take[eds,n]]]/Total[Take[eds,n]],{n,502}],IntegerQ]] (* Harvey P. Dale, Nov 29 2011 *)

A019519 Concatenate odd numbers.

Original entry on oeis.org

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

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

A108728 Number of distinct prime divisors of concatenated even numbers.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 2, 3, 6, 5, 3, 4, 6, 4, 5, 4, 3, 5, 4, 7, 6, 5, 8, 3, 7, 4, 4, 8, 5, 7, 4, 4, 8, 5, 7, 6, 4, 8, 8, 7, 5, 3, 7, 4, 6, 10, 11, 5, 4, 10, 6, 5, 6, 5, 5, 4, 9, 4, 8, 9, 6, 5, 8, 4, 12, 5, 4, 8, 10, 5, 9, 7, 6, 8, 8, 5, 10, 5, 9, 7, 5, 5, 8, 3, 11, 6, 6, 7, 9, 10

Offset: 1

Parthasarathy Nambi, Jun 21 2005

			2 has 1 distinct prime divisors, so a(1) = 1.
24 has 2 distinct prime divisors, so a(2) = 2.
246 has 3 distinct prime divisors, so a(3) = 3.

Apurva Rai and Michael S. Branicky, Prime Factorizations of OEIS A019520.

Cf. A001221, A019520, A105388 (number of divisors).

a[n_] := PrimeNu @ FromDigits @ Flatten[IntegerDigits /@ (2*Range[n])]; Array[a, 30] (* Amiram Eldar, Jan 27 2020 *)

a(n) = A001221(A019520(n)). - Amiram Eldar, Jan 27 2020

a(43)-a(52) from Amiram Eldar, Jan 27 2020
a(53)-a(54) from Jinyuan Wang, Jun 27 2020
a(55)-a(69) from Apurva Rai and Michael S. Branicky, Aug 16 2020
a(70)-a(71) from Max Alekseyev, Mar 21 2023
a(72)-a(89) from Tyler Busby, Mar 23 2023
a(90) from Tyler Busby, Apr 20 2024

A210734 Primes p such that p + 1 or p - 1 is a concatenation of successive even numbers starting from 2.

Original entry on oeis.org

3, 23, 2467, 246809, 246811, 24681012141619, 24681012141618202224262830323436384041, 24681012141618202224262830323436384042444649

Offset: 1

Arkadiusz Wesolowski, May 10 2012

a(9) has 625 digits, a(10) has 1476 digits, a(11) has 5048 digits, a(12) has 39024 digits.

Arkadiusz Wesolowski, Table of n, a(n) for n = 1..10
G. L. Honaker, Jr. and Chris Caldwell, 24681...50451 (625-digits)
Henri & Renaud Lifchitz, PRP Records

Cf. A019520.

lst = {}; c = 0; Do[c = c*10^IntegerLength[n] + n; a = c - 1; If[PrimeQ[a], AppendTo[lst, a]]; b = c + 1; If[PrimeQ[b], AppendTo[lst, b]], {n, 2, 48, 2}]; lst

A105388 Number of divisors of concatenated even numbers.

Original entry on oeis.org

2, 8, 8, 6, 32, 12, 4, 48, 96, 48, 8, 40, 64, 24, 32, 32, 12, 96, 16, 448, 64, 48, 256, 16, 192, 36, 24, 640, 32, 192, 16, 32, 256, 72, 256, 288, 16, 384, 256, 256, 32, 12, 128, 60, 160, 1536, 2048, 64, 16, 2304, 64, 64, 96, 72, 32, 48, 512, 24, 256, 1536, 64, 72, 768, 40, 4096, 48, 16, 512, 1024, 48, 1024

Offset: 1

Parthasarathy Nambi, Apr 30 2005

			The number of divisors of 24 is 8 - which is the second term.
The number of divisors of 246 is 8 - which is the third term.
The number of divisors of 2468 is 6 - which is the fourth term.

Max Alekseyev, Table of n, a(n) for n = 1..90 (terms 88..89 from Tyler Busby).

Cf. A019520, A108728 (number of distinct prime divisors).

Table[DivisorSigma[0,FromDigits[Flatten[IntegerDigits/@(2Range[n])]]],{n,30}] (* Harvey P. Dale, Dec 10 2016 *)

More terms from Harvey P. Dale, Dec 10 2016
a(30)-a(69) from Michael S. Branicky, Feb 08 2021 (computed using prime factorizations linked in A108728)
Offset corrected and a(70)-a(71) added by Max Alekseyev, Mar 21 2023

A349960 Values taken by the function A067095 in the order of their appearance.

Original entry on oeis.org

2, 1, 18, 181, 1817, 18175, 181757, 1817571, 18175719, 181757197, 1817571972, 18175719727, 181757197277, 1817571972772, 18175719727727, 181757197277277, 1817571972772779, 18175719727727795, 181757197277277957, 1817571972772779572, 18175719727727795720, 181757197277277957202

Offset: 1

Bernard Schott, Dec 07 2021

a(2) < a(1), but thereafter this function increases monotonically without limit (see Krusemeyer reference).
The record values > 2 of A067095(m) occur when m = 5, 50, 500, 5000, .... This happens precisely when the corresponding numerator A019520(m) goes from 2/4/6/8/10/12/....../999...98 to 2/4/6/8/10/12/....../999...98/1000...00, where here / means concatenation.
If a(n) is a k-digit number (k = A055642(a(n))), then 1.8 * 10^(k-1) < a(n) < 1.9 * 10^(k-1).
If we consider the sequence u(n) = a(n)/10^(k-1) where k = length(a(n)); we have u(n) is increasing with an upper bound 1.9; so, this sequence u(n) is convergent and, conjecture, this limit = 1.81757197277277957... found by Giorgos Kalogeropoulos; now, from this limit, it is possible to get the successive terms of this sequence here.

			Floor(A019520(5)/A019519(5)) = floor(246810/13579) = floor(18.175859...) = 18, hence, 18 is a term.

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

Chai Wah Wu, Table of n, a(n) for n = 1..1001

Cf. A019519, A019520, A055642, A067095.

terms=5; f[i_]:=FromDigits@Flatten[IntegerDigits/@i];
k[q_]:=f[Range[2,2q,2]]/f[Range[1,2q,2]];
DeleteDuplicates@Table[Floor[k@n],{n,10^(terms-2)/2}] (* Giorgos Kalogeropoulos, Dec 10 2021 *)

def A349960(n): return 3-n if n <= 2 else int("".join(str(d) for d in range(2,10**(n-2)+1,2)))//int("".join(str(d) for d in range(1,10**(n-2),2))) # Chai Wah Wu, Dec 10 2021
from itertools import count
def A349960(n): # a more efficient implementation
    if n <= 2:
        return 3-n
    a, b = '', ''
    for i in count(1,2):
        a += str(i)
        b += str(i+1)
        ai, bi = int(a), int(b)
        if len(a)+n-2 == len(b): return bi//ai
        m = 10**(n-2-len(b)+len(a))
        lb = bi*m//(ai+1)
        ub = (bi+1)*m//ai
        if lb == ub: return lb # Chai Wah Wu, Dec 10 2021

a(n) = floor(k((n + 6)/2)*10^(n - 1 - ceiling(log_10(k((n + 6)/2))))) for k(n) = A019520(n)/A019519(n) and n >= 2 (conjectured). - Giorgos Kalogeropoulos, Dec 10 2021

a(5)-a(7) from Michel Marcus, Dec 07 2021
a(8)-a(9) from Martin Ehrenstein, Dec 10 2021
a(10)-a(22) from Chai Wah Wu, Dec 10 2021

A333183 Number of digits in concatenation of first n positive even integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154

Offset: 1

Alexander Goebel, Mar 10 2020

Connected with A019520 and A038396, similar to how A058183 applies to both A007908 and A000422 to count the digits in them, as the order of the digits does not matter (2468 returns the same result as 8642).

			For example, a(5) = 6 because 246810 (the concatenation of the first five positive even integers) has six digits.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A058183, A007908, A000422, A019520, A038396.

L:= [1$4, seq(i $(9/2*10^(i-1),i=2..3)]:
ListTools:-PartialSums(L); # Robert Israel, Apr 05 2020

Accumulate[IntegerLength[2Range[80]]] (* Alonso del Arte, Mar 12 2020, after Harvey P. Dale *)

a(n) = sum(k=1, n, #Str(2*k)); \\ Michel Marcus, Apr 02 2020

(1 to 80).map{n: Int => new java.math.BigInteger(((2 to (2 * n) by 2).map(.toString)).mkString("")).toString.length} // _Alonso del Arte, Mar 12 2020

a(n) = A058183(n) - Sum_{1..A058183(n)} A000035(A058183(n)).

a(n) = Sum_{i=1..n} (1+floor(log_10(2*i))). - Robert Israel, Apr 05 2020

A286661 Primes of form A038396(n) - 1 or A038396(n) + 1.

Original entry on oeis.org

3, 41, 43, 641, 643, 8641, 108643, 18161412108641, 525048464442403836343230282624222018161412108643, 646260585654525048464442403836343230282624222018161412108641

Offset: 1

XU Pingya, May 12 2017

a(11) = A038396(42) + 1 = 84...43, a(12) = A038396(54) + 1 = 108...43;
a(13) = A038396(185) + 1 = 370...43, a(14) = A038396(199) - 1 = 398...41;
a(15) = A038396(224) + 1 = 448...43, a(16) = A038396(248) - 1 = 496...41;
a(17) = A038396(346) - 1 = 692...41, a(18) = A038396(947) - 1 = 1894...41.
a(19) (if it exists) will be more than  A038396(3000).
a(2) and a(3) are a pair of twin primes, a(4) and a(5) also.

Eric Weisstein's World of Mathematics, Consecutive Number Sequences

Cf. A038396, A019520, A210734.

Select[#, PrimeQ] &@ Flatten@ Table[{# - 1, # + 1} &@ FromDigits@ Flatten@ Reverse@ Take[#, n], {n, Length@ #}] &@ Array[IntegerDigits[2 #] &, 40] (* Michael De Vlieger, May 14 2017 *)

cp's OEIS Frontend

A072724 Integers which are exactly the concatenation of the first m even numbers (A019520) divided by their sum (A002378 = m^2+m).

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A019519 Concatenate odd 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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A108728 Number of distinct prime divisors of concatenated even numbers.

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A210734 Primes p such that p + 1 or p - 1 is a concatenation of successive even numbers starting from 2.

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Mathematica

A105388 Number of divisors of concatenated even numbers.

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A349960 Values taken by the function A067095 in the order of their appearance.

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