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.

A067091 Floor(X/Y) where X = concatenation of the (n+1)-st even number through the (2n)-th even number and Y = concatenation of first n even numbers.

Original entry on oeis.org

2, 2, 329, 4101, 4919, 5737, 6556, 7374, 8193, 9012, 9830, 10649, 11467, 12286, 13104, 13923, 14741, 15560, 16378, 17197, 18015, 18834, 19652, 20471, 212899, 22108437, 2292696195, 237454867452, 24564011532104, 2538253631893694
Offset: 1

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

			a(4) = floor(10121416/2468) = floor(4101.05996758508914100486223662885) = 4101.
a(7) = floor(16182022242628/2468101214) = floor(6556.4662222268166673) = 6556.

Links

Crossrefs

Cf. A067088, A067089, A067090.

Programs

  • Mathematica
    z[n_] := Block[{a = "", m = n}, While[ Length[m] > 0, a = StringJoin[a, ToString[m[[1]]]]; m = Drop[m, 1]]; ToExpression[a]]; Table[ Floor[ z[Table[2i, {i, n + 1, 2n}]] / z[ Table[2i, {i, 1, n}]]], {n, 1, 30}]

Extensions

Edited by N. J. A. Sloane and Robert G. Wilson v, Jun 14 2002

A067092 a(n) = floor(X/Y) where X = concatenation in decreasing order of (2n)-th even number to (n+1)-th even number and Y = that of first n even numbers in increasing order.

Original entry on oeis.org

2, 3, 49, 6540, 8176, 9814, 11451, 13088, 14725, 16362, 17999, 19636, 21273, 22910, 24547, 26184, 27821, 29458, 31095, 32732, 34369, 36006, 37643, 39280, 40917, 4217902, 438013252, 45423626657, 4704592803725, 486682294165577
Offset: 0

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

			a(3) = floor[12108/246] = 49. a(8) = floor [ 3230282624222018/246810121416] = floor[13088.128662184629278356029087656] = 13088.

Links

Crossrefs

Cf. A067088, A067089, A067090, A067091.

Programs

  • Mathematica
    Table[Floor[FromDigits[Flatten[IntegerDigits/@Range[4n,2n+2,-2]]]/ FromDigits[ Flatten[IntegerDigits/@Range[2,2n,2]]]],{n,30}] (* Harvey P. Dale, Mar 11 2019 *)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 01 2003
Edited by Charles R Greathouse IV, Apr 23 2010

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

A067093 Floor[X/Y] where X = concatenation of (n+1)-st odd number through the 2n-th odd number and Y = concatenation of first n odd numbers.

Original entry on oeis.org

3, 4, 58, 6714, 81975, 96852, 111729, 126607, 141484, 156361, 171238, 186116, 200993, 215870, 230748, 245625, 260502, 275379, 290257, 305134, 320011, 334889, 349766, 364643, 379520, 39439811, 4092753962, 424152673322, 43902995043838
Offset: 1

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

			a(4) = floor[9111315/1357] = floor[6714.30729550478997789240972733972] = 6714.

Crossrefs

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

Programs

  • Mathematica
    f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2n + 2k -1]]; y = StringJoin[y, ToString[2k - 1]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 32} ]

Extensions

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

A067094 Floor[X/Y] where X = concatenation in decreasing order of (n+1)-st odd number through the 2n-th odd number and Y = concatenation in increasing order of first n odd numbers.

Original entry on oeis.org

3, 5, 8, 1115, 141185, 170938, 200692, 230447, 260202, 289956, 319711, 349465, 379220, 408974, 438729, 468483, 498238, 527993, 557747, 587502, 617256, 647011, 676765, 706520, 736274, 7592691, 788748988, 81823548335, 8477219785398
Offset: 1

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

			a(4)= floor[1513119/1357] =floor[1115.047162859248341930729550479] = 1115.

Links

Crossrefs

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

Programs

  • Maple
    f:= proc(n) local k; floor(parse(cat(seq(2*k-1,k=2*n .. n+1,-1)))/parse(cat(seq(2*k-1,k=1..n)))) end proc:
map(f, [$1..50]); # Robert Israel, Nov 06 2024
  • Mathematica
    f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[ToString[2n + 2k - 1], x]; y = StringJoin[y, ToString[2k - 1]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]/10]] ); Table[ f[n], {n, 1, 32} ]

Extensions

More terms from Robert G. Wilson v, Jan 09 2002
Definition corrected by Robert Israel, Nov 06 2024

A067090 Floor(X/Y) where X = concatenation of (2n), (2n-1), ... down to n+1 and Y = concatenation of 1,2,3,... up to n.

Original entry on oeis.org

2, 3, 5, 7, 8, 981, 114462, 13082645, 1471900839, 1635537203, 1799173568, 1962809933, 2126446298, 2290082663, 2453719028, 2617355393, 2780991758, 2944628123, 3108264488, 3271900853, 3435537218, 3599173583, 3762809948
Offset: 1

Views

Author

Amarnath Murthy, Jan 07 2002

Keywords

Examples

			a(4) = floor(8765/1234) = 7.
a(6) = floor(121110987/123456) = floor(981.00527313374805598755832037325) = 981.

Crossrefs

Cf. A067088, A067089.

Programs

  • Mathematica
    f[n_] := (k = 1; x = y = "0"; While[k < n + 1, x = StringJoin[x, ToString[2n - k + 1]]; y = StringJoin[y, ToString[2k - 1]]; k++ ]; Return[ Floor[ ToExpression[x] / ToExpression[y]]] ); Table[ f[n], {n, 1, 25} ]
Table[Floor[FromDigits[Flatten[IntegerDigits/@(Range[2n,n+1,-1])]]/FromDigits[Flatten[IntegerDigits/@(Range[n])]]],{n,25}] (* Harvey P. Dale, Jul 24 2011 *)

Extensions

More terms from Robert G. Wilson v, Jan 09 2002
Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 14 2007
Offset corrected by Sean A. Irvine, Dec 02 2023

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
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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