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.

A075000 Smallest number such that n*a(n) is a concatenation of n consecutive integers; or 0 if no such number exists.

Original entry on oeis.org

1, 6, 41, 864, 2469, 20576, 493827, 7098637639, 13717421, 1234567891, 82737383012865106529, 10288065758426, 3513762316247164732, 563643651522439401227280, 8230452606740808761
Offset: 1

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Comments

Conjecture: For every n there exists a nonzero a(n).

Examples

			a(11) = 82737383012865106529 as 11*82737383012865106529 = 910111213141516171819 is the concatenation of 11 numbers from 9 to 19.

Links

Crossrefs

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086, A074999, A075001.

Programs

  • Mathematica
    f[ n_ ] := Block[ {id = Range@n}, While[ k = FromDigits@ Flatten@ IntegerDigits@ id/n; !IntegerQ@k, id++ ]; k ]; Array[ f, 16 ] (* Robert G. Wilson v, Oct 19 2007 *)

Formula

a(n) = A077306(n)/n. - Amarnath Murthy, Nov 03 2002

Extensions

More terms from Rick L. Shepherd, Sep 03 2002

A074996 Floor of concatenation of n, n+1, n+2, n+3, n+4, n+5 divided by 6.

Original entry on oeis.org

2057, 20576, 39094, 57613, 76131, 946485, 11315168, 131516852, 1485018535, 15168520219, 16852021902, 18535523586, 20219025269, 21902526953, 23586028636, 25269530320, 26953032003, 28636533687, 30320035370
Offset: 0

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Crossrefs

Cf. A074991, A074992, A074993, A074994, A074995, A074997, A073086, A074999.

Programs

  • Mathematica
    Table[Floor[FromDigits[Flatten[IntegerDigits[Range[n,n+5]]]]/6],{n,0,18}] (* Jayanta Basu, May 22 2013 *)

Extensions

Edited by Dean Hickerson, Nov 03 2002

A073086 Floor[concatenation of eight consecutive numbers from n to n+7 divided by 8].

Original entry on oeis.org

154320, 1543209, 2932098, 43209863, 570986376, 7098637639, 84863763901, 986376390164, 11137639016426, 113763901642689, 126390164268952, 139016426895214, 151642689521477, 164268952147740, 176895214774002
Offset: 0

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Crossrefs

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A036377, A074999.

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 18 2003

A074993 a(n) = floor((concatenation of n, n+1)/2).

Original entry on oeis.org

0, 6, 11, 17, 22, 28, 33, 39, 44, 455, 505, 556, 606, 657, 707, 758, 808, 859, 909, 960, 1010, 1061, 1111, 1162, 1212, 1263, 1313, 1364, 1414, 1465, 1515, 1566, 1616, 1667, 1717, 1768, 1818, 1869, 1919, 1970, 2020, 2071, 2121, 2172, 2222, 2273, 2323, 2374
Offset: 0

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Comments

The first differences follow a pattern. Odd-indexed terms and even-indexed terms form separate A.P.s with the same common difference for all n except n = 10^k -1. The corresponding common differences are the repunits = (10^(d+1)-1)/9 where d = the number of digits in n.

Crossrefs

Cf. A001704, A074991, A074992, A074994, A074995, A074996, A074997, A073086, A074999, A127421.

Programs

  • Mathematica
    cc[n_]:=Floor[FromDigits[Join[IntegerDigits[n],IntegerDigits[n+1]]]/2]; Array[cc,40,0] (* Harvey P. Dale, Nov 11 2011 *)

A074994 Floor of concatenation of n, n+1, n+2, n+3 divided by 4.

Original entry on oeis.org

30, 308, 586, 864, 1141, 1419, 1697, 19727, 222752, 2275278, 2527803, 2780328, 3032853, 3285379, 3537904, 3790429, 4042954, 4295480, 4548005, 4800530, 5053055, 5305581, 5558106, 5810631, 6063156, 6315682, 6568207, 6820732
Offset: 0

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Examples

			a(7) = floor(78910/4) = 19727.

Crossrefs

Cf. A074991, A074992, A074993, A074995, A074996, A074997, A073086, A074999.

Extensions

Edited by Dean Hickerson, Nov 03 2002

A074995 Floor of concatenation of n, n+1, n+2, n+3, n+4 divided by 5.

Original entry on oeis.org

246, 2469, 4691, 6913, 9135, 11357, 135782, 1578202, 17820222, 182022242, 202224262, 222426283, 242628303, 262830323, 283032343, 303234363, 323436384, 343638404, 363840424, 384042444, 404244464, 424446485, 444648505, 464850525
Offset: 0

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Crossrefs

Cf. A074991, A074992, A074993, A074994, A074996, A074997, A073086, A074999.

Extensions

Edited by Dean Hickerson, Nov 03 2002

A036377 Floor[concatenation of seven consecutive numbers from n to n+6 divided by 7].

Original entry on oeis.org

17636, 176366, 335096, 493827, 6525558, 81127287, 969871587, 11272873030, 127287303044, 1300158875916, 1444459020216, 1588759164516, 1733059308816, 1877359453117, 2021659597417, 2165959741717, 2310259886017, 2454560030317
Offset: 0

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Crossrefs

Cf. A074991, A074992, A074993, A074994, A074995, A074996, A073086, A074999.

Programs

  • Mathematica
    Floor[#/7]&/@(FromDigits[Flatten[IntegerDigits/@#]]&/@Partition[Range[0,25],7,1])  (* Harvey P. Dale, Feb 03 2011 *)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Mar 23 2003

A075001 Smallest k such that the concatenation of n consecutive numbers starting with k (from k to n+k-1) is a multiple of n; or 0 if no such number exists.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 3, 5, 1, 1, 9, 1, 4, 7, 1, 5, 23, 1, 14, 1, 9, 9, 13, 5, 1, 21, 1, 13, 12, 1, 36, 21, 9, 3, 41, 1, 34, 33, 9, 21, 12, 9, 33, 9, 1, 13, 28, 5, 48, 1, 23, 21, 3, 1, 11, 13, 14, 41, 28, 1, 114, 115, 9, 41, 21, 9, 23, 69, 1, 61, 73, 5, 14, 43, 1, 145, 13, 9, 127, 41, 9, 95
Offset: 1

Views

Author

Amarnath Murthy, Aug 31 2002

Keywords

Comments

Conjecture: For every n there exists a k.
First occurrence of k where a(n)=k: 1, 103, 4, 13, 8, 105, 14, 87, 11, 699, 55, 29, 23, 19, 114, 261, 102, 97, 178, 219, 26, 121, 17, 151, 92, ..., . - Robert G. Wilson v
a(n)=1 iff n is in A029455. - Robert G. Wilson v
Increasing a(n)'s: 1, 3, 5, 9, 23, 36, 41, 48, 114, 115, 145, 166, 175, 221, 251, ..., at n = 1, 4, 8, 11, 17, 31, 35, 49, 61, 62, 76, 85, 122, 133, 170, 179, 217, 229, ..., . - Robert G. Wilson v

Examples

			a(11) = 9 as 910111213141516171819 the concatenation of 11 numbers from 9 to 19 is divisible by 11 (11*82737383012865106529).

Links

Crossrefs

Cf. A058183, A074991, A074992, A074993, A074994, A074995, A074996, A036377, A073086, A074999, A075000, A077306, A075000.
Cf. A029455 (indices of 1's).

Programs

  • Mathematica
    f[n_] := Block[{c = 1, id = Range@n}, While[k = FromDigits@Flatten@IntegerDigits@id/n; ! IntegerQ@k, id++; c++ ]; c]; Array[f, 82] (* Robert G. Wilson v, Oct 20 2007 *)
  • PARI
    /* The following program assumes the conjecture is true. */ /* It has found nonzero a(n) for n up to 500. */ {for(n=1,500, k=0; until(s%n==0,k++; s=0; for(m=k,k+n-1, s=s*(10^length(Str(m)))+m)); print1(k,","))}
  • PARI
    a(n) = {my(ld = 1, hd = n, qd, m = Mod(1, n), pow10, qdn = #digits(n), t=log(10*n+.5)\log(10)); qd = n*t+t-10^t\9; pow10 = Mod(10, n)^(qd-1); for(i = 2, n, m = m * Mod(10, n)^#digits(i) + i; ); while(1, if(lift(m) == 0, return(ld)); m -= ld * pow10; hd++; m = m * Mod(10, n)^#digits(hd) + hd; ld++; pow10*=10^(#digits(hd) - #digits(ld)); ) } \\ David A. Corneth, Aug 23 2020

Extensions

More terms from Rick L. Shepherd, Sep 03 2002

A075008 Floor[ concatenation of 7 numbers from n+6 to n in that order divided by 7].

Original entry on oeis.org

934744, 1093474, 1252204, 1410934, 1569664, 15871252, 173015696, 1874444426, 20187444442, 216304458729, 2307344731587, 2451644875887, 2595945020187, 2740245164487, 2884545308787, 3028845453087, 3173145597388, 3317445741688
Offset: 0

Views

Author

Amarnath Murthy, Sep 01 2002

Keywords

Links

Crossrefs

Cf. A074991 to A075000, A075003 to A075007, A074999, A075010.

Programs

  • Mathematica
    Table[Floor[FromDigits[Flatten[IntegerDigits/@Range[n+6,n,-1]]]/7],{n,0,20}] (* Harvey P. Dale, May 20 2021 *)

Extensions

More terms from Antonio G. Astudillo (afg_astudillo(AT)lycos.com), Apr 18 2003
Showing 1-9 of 9 results.

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

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