cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Comments

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

Examples

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

References

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

Crossrefs

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094.
Cf. A019519, A019520, A349960.

Programs

  • Mathematica
    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 *)
  • PARI
    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

Formula

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

Extensions

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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Comments

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

Examples

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

Crossrefs

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094, A067095.
floor[A019520(n)/A019519(n)]

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

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

Crossrefs

Cf. A067088, A067089, A067090, A067091, A067092, A067093, A067094, A067095, A067096.
floor[A045507(n)/A019519(n)]

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

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

Crossrefs

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

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

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

Crossrefs

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099.
floor[A019518(n)/A007908(n)]

Programs

  • Mathematica
    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 *)

Extensions

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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

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

Crossrefs

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101.
floor[A019521(n)/A007908(n)]

Programs

  • Mathematica
    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} ]

Extensions

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

Views

Author

Robert G. Wilson v, Jan 09 2002

Keywords

Comments

a(n) -> 148038942651147505942387547594667814093751032610233441970375...

Examples

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

Crossrefs

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

Programs

  • Maple
    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
  • Mathematica
    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 *)
  • PARI
    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

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

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

Links

Crossrefs

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

Programs

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

Extensions

Corrected and extended by Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003
Edited by Charles R Greathouse IV, Apr 27 2010

A067105 a(n) = floor[ X/Y], where X = concatenation of k^k from 1 up to n^n and Y = concatenation of 1, ..., n.

Original entry on oeis.org

1, 1, 11, 1156, 1156141, 11560850121, 1156078457100065, 11560777079611640798854, 1156077623683098402586161358986, 1156077622746675519639905953267558458549
Offset: 1

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

			a(5) = floor [14272563125/12345] = floor[1156141.20089104900769542324827866] = 1156141.

Crossrefs

Cf. A067091, A067092, A067093, A067094, A067095, A067096, A067097, A067098, A067099, A067100, A067101, A067102, A067103, A067104.

Programs

  • Mathematica
    fxy[n_]:=Module[{num=FromDigits[Flatten[IntegerDigits/@(Table[x^x,{x,n}])]], den=FromDigits[Flatten[IntegerDigits/@Range[n]]]},Floor[num/den]]; Array[ fxy,10] (* Harvey P. Dale, Mar 21 2013 *)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003
Edited by Charles R Greathouse IV, Apr 28 2010
Showing 1-9 of 9 results.

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