A067095 -id:A067095 - cp's OEIS frontend

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

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

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

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

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

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

A067098 Floor[X/Y] where X = concatenation in increasing order of first n powers of 3 and Y = that of first n natural numbers.

Original entry on oeis.org

3, 3, 31, 318, 31817, 3181548, 3181530396, 3181528335091, 31815281031585777, 31815281005815399552, 318152810055319253966698, 3181528100552883295133046294, 318152810055287994498392866979206

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(4)= floor [ 392781/1234] = floor[318.299027552674230145867098865478] = 318.

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094, A067095, A067096, A067097.

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[3^k]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 15} ]
Table[Floor[FromDigits[Flatten[IntegerDigits/@(3^Range[n])]]/ FromDigits[ Flatten[IntegerDigits/@Range[n]]]],{n,15}] (* Harvey P. Dale, Mar 10 2019 *)

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

A067101 Floor[ X/Y], where X = concatenation of the primes and Y = concatenation of natural numbers.

Original entry on oeis.org

2, 1, 1, 1, 19, 190, 1909, 19092, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 190926, 1909260, 19092601, 190926018, 1909260182, 19092601827, 190926018273

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(5) = floor [235711/12345]=floor[19.093641150...] = 19.

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099.

floor[A019518(n)/A007908(n)]

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[Prime[k]]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 25} ]
nn=40;With[{prs=Prime[Range[nn]],nats=Range[nn]},Table[Floor[FromDigits[ Flatten[IntegerDigits/@Take[prs,n]]]/FromDigits[Flatten[IntegerDigits /@Take[nats,n]]]],{n,nn}]] (* Harvey P. Dale, Mar 24 2012 *)

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

A067102 Floor[ X/Y] where X = concatenation of the squares and Y = concatenation of natural numbers.

Original entry on oeis.org

1, 1, 1, 12, 120, 1208, 12082, 120821, 1208216, 12082165, 120821655, 1208216555, 12082165556, 120821655562, 1208216555626, 12082165556267, 120821655562672, 1208216555626728, 12082165556267282, 120821655562672822

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(5) = floor [1491625/12345]=floor[] = floor[120.828270554880518428513568246254]=120.

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101.

floor[A019521(n)/A007908(n)]

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[k^2]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 20} ]

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

A067103 a(n) = floor(X/Y), where X = concatenation of cubes and Y = concatenation of natural numbers.

Original entry on oeis.org

1, 1, 14, 148, 14804, 1480398, 148039049, 14803895356, 1480389427723, 148038942652481, 14803894265116205, 1480389426511476635, 148038942651147507639, 14803894265114750596056, 1480389426511475059425814, 148038942651147505942389607, 14803894265114750594238756940

Offset: 1

Robert G. Wilson v, Jan 09 2002

a(n) -> 148038942651147505942387547594667814093751032610233441970375...

			a(6) = floor(182764125216/123456) = floor(1480398.888802...) = 1480398.

Cf. A000578, A007908, A019522.
Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101, A067102.
See also A066700.

a:= n-> floor(parse(cat(i^3$i=1..n))/parse(cat($1..n))):
seq(a(n), n=1..17);  # Alois P. Heinz, May 25 2022

f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[k^3]]; y = StringJoin[y, ToString[k]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 20} ]
nn=20;With[{c=Table[IntegerDigits[n^3],{n,nn}],s=Table[IntegerDigits[n],{n,nn}]}, Table[Floor[FromDigits[Flatten[Take[c,i]]]/FromDigits[Flatten[Take[s,i]]]],{i,nn}]] (* Harvey P. Dale, Feb 10 2013 *)

c1(n) = my(s=""); for(k=1, n, s=Str(s, k)); eval(s); \\ A007908
c3(n) = my(s=""); for(k=1, n, s=Str(s, k^3)); eval(s); \\ A019522
a(n) = c3(n)\c1(n); \\ Michel Marcus, May 25 2022

A067104 a(n) = floor[ X/Y], where X = concatenation of first n factorials and Y = concatenation of first n natural numbers.

Original entry on oeis.org

1, 1, 1, 10, 1022, 102256, 102255452, 1022553862210, 102255378766606673, 10225537868377981588347, 10225537868286872045185666318, 102255378682858781228966381713174081, 10225537868285867355405173700779791589867289

Offset: 1

Amarnath Murthy, Jan 07 2002

			a(5) = floor [12624120/12345] = floor[1022.60996354799513973268529769137] = 1022.

Harvey P. Dale, Table of n, a(n) for n = 1..44

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101, A067102, A067103.

Table[Floor[FromDigits[Flatten[IntegerDigits/@(Range[n]!)]]/FromDigits[ Flatten[IntegerDigits/@Range[n]]]],{n,15}] (* Harvey P. Dale, Jun 09 2020 *)

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003

Edited by Charles R Greathouse IV, Apr 27 2010

cp's OEIS Frontend

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

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A019519 Concatenate odd numbers.

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A019520 a(n) is the concatenation of the first n positive even 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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A067097 Floor[X/Y] where X = concatenation in increasing order of first n powers of 2 and Y = that of first n natural numbers.

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A067098 Floor[X/Y] where X = concatenation in increasing order of first n powers of 3 and Y = that of first n natural numbers.

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A067101 Floor[ X/Y], where X = concatenation of the primes and Y = concatenation of natural numbers.

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A067102 Floor[ X/Y] where X = concatenation of the squares and Y = concatenation of natural numbers.

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