A074992 -id:A074992 - cp's OEIS frontend

A284691 Numbers of the form (10^c-1)*the product any two (not necessarily distinct) terms of A074992.

9, 99, 333, 999, 3663, 9999, 12321, 30303, 36963, 99999, 135531, 333333, 369963, 999999, 1121211, 1367631, 3003003, 3363633, 3699963, 9999999, 12333321, 13688631, 33033033, 33666633, 36999963, 99999999, 102030201, 111111111, 124454421, 136898631, 300030003

Offset: 1

Ahmad J. Masad, Apr 01 2017

Conjecture 1: all terms are palindromic in base 10.

Conjecture 2: the sequence A074992 is the maximally dense sequence with this palindromic products property.

			a(3) = 37*9 = 333, with respect to strictly increasing ordering.

Cf. A002283, A074992, A237424.

f[n_] := f[n] = (10^(2 n) + 10^n + 1)/3; c[n_] := 10^n - 1; mx = 10^10; i=1; Union@ Reap[ While[c[i] <= mx, j=0; While[c[i] f[j] <= mx, k=0; While[k <= j && (v = c[i] f[j] f[k]) <= mx, Sow@v; k++]; j++]; i++]][[2, 1]] (* Giovanni Resta, Apr 01 2017 *)

a(13)-a(31) from Giovanni Resta, Apr 01 2017

A074991 Concatenation of n, n+1, n+2 divided by 3.

Original entry on oeis.org

4, 41, 78, 115, 152, 189, 226, 263, 2970, 30337, 33704, 37071, 40438, 43805, 47172, 50539, 53906, 57273, 60640, 64007, 67374, 70741, 74108, 77475, 80842, 84209, 87576, 90943, 94310, 97677, 101044, 104411, 107778, 111145, 114512

Offset: 0

Amarnath Murthy, Aug 31 2002

If all the three numbers have the same number of digits then the terms are in A.P. with a common difference that follows a pattern with increase in number of digits. E.g. for n = 1 to 7 the common difference, c.d. = 37. For n = 10 to 97 c.d. = 3367, For three digit numbers from 100 to 997 it is 333667 etc.

			a(4) = 456/3 = 152.

Cf. A074992.

Equals (1/3) A001703.

read("transforms") ;
A074991 := proc(n)
        digcatL([n,n+1,n+2]) ;
        %/3 ;
end proc:
seq(A074991(n),n=0..50) ; # R. J. Mathar, Oct 04 2011

f[n_] := FromDigits@ Flatten@ IntegerDigits[{n, n + 1, n + 2}]/3; Array[f, 26, 0] (* Robert G. Wilson v *)

Incorrect conjecture deleted by Colin Barker, Sep 26 2013

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

Amarnath Murthy, Aug 31 2002

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

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

Robert G. Wilson v, Table of n, a(n) for n = 1..100

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

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 *)

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

More terms from Rick L. Shepherd, Sep 03 2002

A066138 a(n) = 10^(2*n) + 10^n + 1.

Original entry on oeis.org

3, 111, 10101, 1001001, 100010001, 10000100001, 1000001000001, 100000010000001, 10000000100000001, 1000000001000000001, 100000000010000000001, 10000000000100000000001, 1000000000001000000000001, 100000000000010000000000001, 10000000000000100000000000001, 1000000000000001000000000000001

Offset: 0

Henry Bottomley, Dec 07 2001

Palindromes whose digit sum is 3.
Essentially the same as A135577. - R. J. Mathar Apr 29 2008
From Peter Bala, Sep 25 2015: (Start)
For n >= 1, the simple continued fraction expansion of sqrt(a(n)) = [10^n; 1, 1, 2/3*(10^n - 1), 1, 1, 2*10^n, ...] has period 6. As n increases, the expansion has the large partial quotients 2/3*(10^n - 1) and 2*10^n.
For n >= 1, the continued fraction expansion of sqrt(a(2*n))/a(n) = [0; 1, 10^n - 1, 1, 1, 1/3*(10^n - 4), 1, 4, 1, 1/3*(10^n - 4), 1, 1, 10^n - 1, 2, 10^n - 1, ...] has pre-period of length 3 and period 12 beginning 1, 1, 1/3*(10^n - 4), ....  As n increases, the expansion has the large partial quotients 10^n - 1 and 1/3*(10^n - 4).
A theorem of Kuzmin in the measure theory of continued fractions says that large partial quotients are the exception in continued fraction expansions.
Empirically, we also see exceptionally large partial quotients in the continued fraction expansions of the m-th root of the numbers a(m*n), for m >= 3. For example, it appears that the continued fraction expansion of a(3*n)^(1/3), for n >= 2, begins [10^(2*n); 3*10^n - 1, 1, 0.5*10^(2*n) - 1, 1.44*10^n - 1, 1, ...]. Cf. A000533, A002283 and A168624. (End)

			From _Peter Bala_, Sep 25 2015: (Start)
Simple continued fraction expansions showing large partial quotients:
a(9)^(1/3) =[1000000; 2999, 1, 499999, 1439, 1, 2582643, 1, 1, 1, 2, 3, 3, ...].
a(20)^(1/4) = [10000000000; 39999999999, 1, 3999999999, 16949152542, 2, 1, 2, 6, 1, 4872106, 3, 9, 2, 3, ...].
a(25)^(1/5) = [10000000000; 4999999999999999, 1, 3333333332, 2, 1, 217391304347825, 2, 2, 1, 1, 1, 2, 1, 23980814, 1, 1, 1, 1, 1, 7, ...]. (End)

Harry J. Smith, Table of n, a(n) for n=0..100
Index entries for linear recurrences with constant coefficients, signature (111,-1110,1000).

Cf. A000533, A002283, A033934, A062397, A074992, A135577, A168624.

[10^(2*n) + 10^n + 1: n in [0..20]]; // Vincenzo Librandi, Sep 27 2015

Table[10^(2 n) + 10^n + 1, {n, 0, 15}] (* Michael De Vlieger, Sep 27 2015 *)
CoefficientList[Series[(3 - 222 x + 1110 x^2)/((1 - 100 x) (1 - 10 x) (1 - x)), {x, 0, 33}], x] (* Vincenzo Librandi, Sep 27 2015 *)

a(n) = { 10^(2*n) + 10^n + 1 } \\ Harry J. Smith, Feb 02 2010

Vec(-3*(370*x^2-74*x+1)/((x-1)*(10*x-1)*(100*x-1)) + O(x^20)) \\ Colin Barker, Sep 27 2015

A168624(n) = a(2*n)/a(n). - Peter Bala, Sep 24 2015
G.f.: (3 - 222*x + 1110*x^2)/((1 - 100*x)*(1 - 10*x)*(1 - x)). - Vincenzo Librandi, Sep 27 2015
From Colin Barker, Sep 27 2015: (Start)
a(n) = 111*a(n-1) - 1110*a(n-2) + 1000*a(n-3) for n > 2.
G.f.: -3*(370*x^2-74*x+1)/((x-1)*(10*x-1)*(100*x-1)). (End)
From Elmo R. Oliveira, Aug 27 2024: (Start)
E.g.f.: exp(x)*(exp(99*x) + exp(9*x) + 1).
a(n) = 3*A074992(n). (End)

Offset changed from 1 to 0 by Harry J. Smith, Feb 02 2010

More terms from Michael De Vlieger, Sep 27 2015

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

Amarnath Murthy, Aug 31 2002

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

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

Edited by Dean Hickerson, Nov 03 2002

A074999 Floor[concatenation of nine consecutive numbers from n to n+8 divided by 9].

Original entry on oeis.org

1371742, 13717421, 260630990, 3840876779, 50754344568, 630990012357, 7543445680146, 87677901347935, 990012357015724, 10112346812683513, 11234681268351302, 12357015724019091, 13479350179686880

Offset: 0

Amarnath Murthy, Aug 31 2002

Harvey P. Dale, Table of n, a(n) for n = 0..1000

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

Table[FromDigits[Flatten[IntegerDigits/@Range[n,n+8]]]/9,{n,0,20}] (* Harvey P. Dale, Feb 19 2018 *)

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

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

Amarnath Murthy, Aug 31 2002

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

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

Amarnath Murthy, Aug 31 2002

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.

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

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

Amarnath Murthy, Aug 31 2002

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

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

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

Amarnath Murthy, Aug 31 2002

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

Edited by Dean Hickerson, Nov 03 2002

cp's OEIS Frontend

A284691 Numbers of the form (10^c-1)*the product any two (not necessarily distinct) terms of A074992.

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A074991 Concatenation of n, n+1, n+2 divided by 3.

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Maple

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

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A066138 a(n) = 10^(2*n) + 10^n + 1.

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

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A074999 Floor[concatenation of nine consecutive numbers from n to n+8 divided by 9].

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

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

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