A009470 -id:A009470 - cp's OEIS frontend

A020338 Doublets: base-10 representation is the juxtaposition of two identical strings.

11, 22, 33, 44, 55, 66, 77, 88, 99, 1010, 1111, 1212, 1313, 1414, 1515, 1616, 1717, 1818, 1919, 2020, 2121, 2222, 2323, 2424, 2525, 2626, 2727, 2828, 2929, 3030, 3131, 3232, 3333, 3434, 3535, 3636, 3737, 3838, 3939, 4040, 4141, 4242, 4343, 4444, 4545, 4646

Offset: 1

David W. Wilson

Reinhard Zumkeller, Table of n, a(n) for n = 1..9999
Eric Weisstein's World of Mathematics, Concatenation

Cf. concatenation of n and k*n: this sequence (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

Cf. A004216, A056524.

Flat(List([1..2],d->List([10^(d-1)..10^d-1],n->(10^d+1)*n))); # Muniru A Asiru, Mar 31 2018

a020338 n = read (ns ++ ns) :: Integer  where ns = show n
-- Reinhard Zumkeller, Jun 07 2015

[Seqint(Intseq(n) cat Intseq(n)): n in [1..46]]; // Bruno Berselli, Mar 20 2013

seq(seq((10^d+1)*n, n = 10^(d-1)..10^d-1),d=1..3); # Robert Israel, Jan 02 2015

nxt[n_]:=Module[{idn=IntegerDigits[n], idn1=IntegerDigits[n]}, FromDigits[Join[idn, idn1]]];Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

a(n) = eval(Str(n,n)); \\ Michel Marcus, Sep 10 2015

[int(str(n)+str(n)) for n in range(1,47)] # Danny Rorabaugh, Oct 10 2015

a(n) = n*10^(A004216(n)+1) + n. - Reinhard Zumkeller, Aug 11 2007
G.f.: 11*x/(1-x)^2 - Sum_{d >= 1} 9*x^(10^d)*(100^d*x-10^d*x-100^d)/(1-x)^2. - Robert Israel, Jan 02 2015
a(n) = n || n. (Where "||" denotes "concatenate". See link "Concatenation".) - Halfdan Skjerning, Apr 01 2018

A019550 a(n) is the concatenation of n and 2n.

Original entry on oeis.org

12, 24, 36, 48, 510, 612, 714, 816, 918, 1020, 1122, 1224, 1326, 1428, 1530, 1632, 1734, 1836, 1938, 2040, 2142, 2244, 2346, 2448, 2550, 2652, 2754, 2856, 2958, 3060, 3162, 3264, 3366, 3468, 3570, 3672, 3774, 3876, 3978, 4080, 4182, 4284, 4386, 4488, 4590

Offset: 1

R. Muller

Concatenation of digits of n and 2*n. - Harvey P. Dale, Sep 13 2011

All terms are divisible by 6. - Robert Israel, Sep 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. concatenation of n and k*n: A020338 (k=1), this sequence (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).
Cf. A235497.
Supersequence of A117304.

[Seqint(Intseq(2*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

seq(n*(10^(1+ilog10(2*n))+2),n=1..100); # Robert Israel, Sep 21 2015

nxt[n_]:=Module[{idn=IntegerDigits[n],idn2=IntegerDigits[2n]}, FromDigits[ Join[ idn,idn2]]]; Array[nxt,40] (* Harvey P. Dale, Sep 13 2011 *)

a(n) = eval(Str(n, 2*n)); \\ Michel Marcus, Sep 21 2015

def a(n): return int(str(n) + str(2*n))
print([a(n) for n in range(1, 46)]) # Michael S. Branicky, Dec 24 2021

From Robert Israel, Sep 21 2015 (Start)
G.f.: (6*(2*x+75*x^5-60*x^6) + 90*Sum_{k>=1} 10^k*x^(5*10^k)*(5*10^k - (5*10^k-1)*x))/(1-x)^2.
a(n+2) - 2*a(n+1) + a(n) = 45*10^(2*k+1) if n = 5*10^k-2, 90*10^k-450*10^(2*k) if n = 5*10^k-1, 0 otherwise. (End)

Offset changed from 0 to 1 by Vincenzo Librandi, Feb 04 2014

A009474 a(n) is the concatenation of n and 9n.

Original entry on oeis.org

19, 218, 327, 436, 545, 654, 763, 872, 981, 1090, 1199, 12108, 13117, 14126, 15135, 16144, 17153, 18162, 19171, 20180, 21189, 22198, 23207, 24216, 25225, 26234, 27243, 28252, 29261, 30270, 31279, 32288, 33297, 34306, 35315, 36324, 37333, 38342, 39351

Offset: 1

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. concatenation of n and k*n: A020338 (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), this sequence (k=9).

[Seqint(Intseq(9*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

nxt[n_]:=Module[{idn=IntegerDigits[n], idn9=IntegerDigits[9n]}, FromDigits[Join[idn, idn9]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

A009440 a(n) is the concatenation of n and 6n.

Original entry on oeis.org

16, 212, 318, 424, 530, 636, 742, 848, 954, 1060, 1166, 1272, 1378, 1484, 1590, 1696, 17102, 18108, 19114, 20120, 21126, 22132, 23138, 24144, 25150, 26156, 27162, 28168, 29174, 30180, 31186, 32192, 33198, 34204, 35210, 36216, 37222, 38228, 39234, 40240

Offset: 1

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. concatenation of n and k*n: A020338 (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), this sequence (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

[Seqint(Intseq(6*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

nxt[n_]:=Module[{idn=IntegerDigits[n], idn6=IntegerDigits[6n]}, FromDigits[Join[idn, idn6]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)
Table[n*10^IntegerLength[6n]+6n,{n,40}] (* Harvey P. Dale, Jul 21 2020 *)

A009441 a(n) is the concatenation of n and 7n.

Original entry on oeis.org

17, 214, 321, 428, 535, 642, 749, 856, 963, 1070, 1177, 1284, 1391, 1498, 15105, 16112, 17119, 18126, 19133, 20140, 21147, 22154, 23161, 24168, 25175, 26182, 27189, 28196, 29203, 30210, 31217, 32224, 33231, 34238, 35245, 36252, 37259, 38266, 39273, 40280

Offset: 1

David W. Wilson

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. concatenation of n and k*n: A020338 (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), this sequence (k=7), A009470 (k=8), A009474 (k=9).

[Seqint(Intseq(7*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

nxt[n_]:=Module[{idn=IntegerDigits[n], idn7=IntegerDigits[7n]}, FromDigits[Join[idn, idn7]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)
Table[n*10^IntegerLength[7n]+7n,{n,40}] (* Harvey P. Dale, Aug 02 2024 *)

A019551 a(n) is the concatenation of n and 3n.

Original entry on oeis.org

13, 26, 39, 412, 515, 618, 721, 824, 927, 1030, 1133, 1236, 1339, 1442, 1545, 1648, 1751, 1854, 1957, 2060, 2163, 2266, 2369, 2472, 2575, 2678, 2781, 2884, 2987, 3090, 3193, 3296, 3399, 34102, 35105, 36108

Offset: 1

R. Muller

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. concatenation of n and k*n: A020338 (k=1), A019550 (k=2), this sequence (k=3), A019552 (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

[Seqint(Intseq(3*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

a:=n->n*10^floor(log10(3*n)+1)+3*n: seq(a(n),n=1..50); # Muniru A Asiru, Jun 23 2018

nxt[n_]:=Module[{idn=IntegerDigits[n], idn3=IntegerDigits[3n]}, FromDigits[Join[idn, idn3]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)
Table[n*10^IntegerLength[3n]+3n,{n,40}] (* Harvey P. Dale, Apr 24 2022 *)

A019552 a(n) is the concatenation of n and 4n.

Original entry on oeis.org

14, 28, 312, 416, 520, 624, 728, 832, 936, 1040, 1144, 1248, 1352, 1456, 1560, 1664, 1768, 1872, 1976, 2080, 2184, 2288, 2392, 2496, 25100, 26104, 27108, 28112, 29116, 30120, 31124, 32128, 33132, 34136, 35140, 36144, 37148, 38152, 39156, 40160, 41164

Offset: 1

R. Muller

a(n) is divisible by 4 for n >= 2. - Michel Marcus, Sep 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. concatenation of n and k*n: A020338 (k=1), A019550 (k=2), A019551 (k=3), this sequence (k=4), A019553 (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

[Seqint(Intseq(4*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

a:=n->n*10^floor(log10(4*n)+1)+4*n: seq(a(n),n=1..50); # Muniru A Asiru, Jun 23 2018

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[4n]]],{n,50}] (* Harvey P. Dale, May 11 2011 *)
nxt[n_]:=Module[{idn=IntegerDigits[n], idn4=IntegerDigits[4n]}, FromDigits[Join[idn, idn4]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

a(n) = eval(Str(n, 4*n)); \\ Michel Marcus, Sep 21 2015

A019553 a(n) is the concatenation of n and 5n.

Original entry on oeis.org

15, 210, 315, 420, 525, 630, 735, 840, 945, 1050, 1155, 1260, 1365, 1470, 1575, 1680, 1785, 1890, 1995, 20100, 21105, 22110, 23115, 24120, 25125, 26130, 27135, 28140, 29145, 30150, 31155, 32160, 33165, 34170, 35175, 36180, 37185, 38190, 39195, 40200

Offset: 1

R. Muller

All terms are divisible by 15. - Michel Marcus, Sep 21 2015

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Sylvester Smith, A Set of Conjectures on Smarandache Sequences, Bulletin of Pure and Applied Sciences, (Bombay, India), Vol. 15 E (No. 1), 1996, pp. 101-107.

Cf. concatenation of n and k*n: A020338 (k=1), A019550 (k=2), A019551 (k=3), A019552 (k=4), this sequence (k=5), A009440 (k=6), A009441 (k=7), A009470 (k=8), A009474 (k=9).

[Seqint(Intseq(5*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

a:=n->n*10^floor(log10(5*n)+1)+5*n: seq(a(n),n=1..50); # Muniru A Asiru, Jun 23 2018

n5n[n_]:=Module[{n5=5n},n*10^IntegerLength[n5]+n5]; Array[n5n,40] (* Harvey P. Dale, Apr 08 2012 *)
nxt[n_]:=Module[{idn=IntegerDigits[n], idn5=IntegerDigits[5n]}, FromDigits[Join[idn, idn5]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

a(n) = eval(Str(n, 5*n)); \\ Michel Marcus, Sep 21 2015

A115549 Numbers k such that the concatenation of k with 8*k gives a square.

Original entry on oeis.org

3, 12, 28, 63, 112, 278, 1112, 2778, 11112, 27778, 111112, 277778, 1111112, 2777778, 4938272, 7716050, 11111112, 12802888, 13151250, 13504288, 13862002, 14224392, 14591458, 14963200, 15339618, 15720712, 16106482, 16496928, 16892050, 17291848, 17696322, 18105472

Offset: 1

Giovanni Resta, Jan 25 2006

If k = 10*R_m + 2, with m >= 1, then the concatenation of k with 8*k equals (30*R_m + 6)^2, so A047855 \ {1,2} is a subsequence. - Bernard Schott, Apr 09 2022
Numbers k such that A009470(k) is a square. - Michel Marcus, Apr 09 2022
The numbers 28, 278, 2778, ..., 2*10^k + 7*(10^k - 1)/9 + 1, ..., k >= 1, are terms, because the concatenation forms the squares 28224 = 168^2, 2782224 = 1668^2, 277822224 = 16668^2, ..., (10^m + 2*(10^m - 1)/3 + 2)^2, m >= 2, ... - Marius A. Burtea, Apr 10 2022

			3_24 = 18^2.
11112_88896 = 33336^2.

Marius A. Burtea, Table of n, a(n) for n = 1..176

Cf. A002275, A047855, A009470, A102567, A106497, A115527-A115556.

[n:n in [1..20000000]|IsSquare(Seqint(Intseq(8*n) cat Intseq(n)))]; // Marius A. Burtea, Apr 10 2022

isok(k) = issquare(eval(Str(k, 8*k))); \\ Michel Marcus, Apr 09 2022

More terms from Marius A. Burtea, Apr 13 2022

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A020338 Doublets: base-10 representation is the juxtaposition of two identical strings.

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A019550 a(n) is the concatenation of n and 2n.

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A009474 a(n) is the concatenation of n and 9n.

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A009440 a(n) is the concatenation of n and 6n.

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A009441 a(n) is the concatenation of n and 7n.

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A019551 a(n) is the concatenation of n and 3n.

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A019552 a(n) is the concatenation of n and 4n.

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