A020338 -id:A020338 - cp's OEIS frontend

A053061 a(n) is the decimal concatenation of n and n^2.

11, 24, 39, 416, 525, 636, 749, 864, 981, 10100, 11121, 12144, 13169, 14196, 15225, 16256, 17289, 18324, 19361, 20400, 21441, 22484, 23529, 24576, 25625, 26676, 27729, 28784, 29841, 30900, 31961, 321024, 331089, 341156, 351225, 361296, 371369, 381444, 391521

Offset: 1

Felice Russo, Feb 25 2000

Felice Russo, A set of new Smarandache functions, sequences and conjectures in number theory, American Research Press 2000

Georg Fischer, Table of n, a(n) for n = 1..10000

Cf. A020338, A055436, A061086, A175605, A253445.

[Seqint(Intseq(n^2) cat Intseq(n)): n in [1..40]]; // Vincenzo Librandi, Jan 03 2015

Table[FromDigits[Join[IntegerDigits[n],IntegerDigits[n^2]]],{n,40}] (* Harvey P. Dale, May 24 2012 *)

def a(n): return int(str(n) + str(n*n))
print([a(n) for n in range(1, 40)]) # Michael S. Branicky, Nov 24 2021

a(n) = n*(10^floor(2*log_10(n) + 1) + n). - Henry Bottomley, May 18 2000

a(n) ~ n^3. - Charles R Greathouse IV, Sep 19 2012

More terms from James Sellers, Feb 28 2000

A009470 a(n) is the concatenation of n and 8n.

Original entry on oeis.org

18, 216, 324, 432, 540, 648, 756, 864, 972, 1080, 1188, 1296, 13104, 14112, 15120, 16128, 17136, 18144, 19152, 20160, 21168, 22176, 23184, 24192, 25200, 26208, 27216, 28224, 29232, 30240, 31248, 32256, 33264, 34272, 35280, 36288, 37296, 38304, 39312

Offset: 1

David W. Wilson

All terms are divisible by 9. - Michel Marcus, Sep 21 2015

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), this sequence (k=8), A009474 (k=9).

[Seqint(Intseq(8*n) cat Intseq(n)): n in [1..50]]; // Vincenzo Librandi, Feb 04 2014

nxt[n_]:=Module[{idn=IntegerDigits[n], idn8=IntegerDigits[8n]}, FromDigits[Join[idn,idn8]]]; Array[nxt, 100] (* Vincenzo Librandi, Feb 04 2014 *)

a(n) = eval(Str(n, 8*n)); \\ Michel Marcus, Sep 21 2015

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

A064001 Odd abundant numbers not divisible by 5.

Original entry on oeis.org

81081, 153153, 171171, 189189, 207207, 223839, 243243, 261261, 279279, 297297, 351351, 459459, 513513, 567567, 621621, 671517, 729729, 742203, 783783, 793611, 812889, 837837, 891891, 908523, 960687, 999999, 1024947, 1054053, 1072071

Offset: 1

Harvey P. Dale, Sep 17 2001

nonn

Or, odd abundant numbers that do not end in 5.
All terms below 2000000 are divisible by 21 (so by 3). Moreover, except for a few, most are divisible by 231. - Labos Elemer, Sep 15 2005 [The least term that is not divisible by 21 is a(908) = 28683369. - Amiram Eldar, Jan 27 2025]
An odd abundant number (see A005231) not divisible by 3 nor 5 must have at least 15 distinct prime factors (e.g., 61#/5#*7^2*11*13*17, where # is primorial) and be >= 67#/5#*77 = A047802(3) ~ 2.0*10^25. -- The smallest non-primitive abundant number (cf. A006038) in this sequence is 7*a(1) = 567567 = a(14). - M. F. Hasler, Jul 27 2016
There are 26 terms less than 10^6 and a surprising fact is that 18 of them are doublets (cf. A020338). - Omar E. Pol, Jan 17 2025
The numbers of terms that do not exceed 10^k, for k = 5, 6, ..., are 1, 26, 290, 3071, 31600, 320948, 3174762, 31693948, ... . Apparently, the asymptotic density of this sequence equals 0.000031... . Therefore, the least term not divisible by 3 that was mentioned above is a(~6*10^20) = 20169691981106018776756331. - Amiram Eldar, Jan 27 2025

David Wells, The Penguin Dictionary of Curious and Interesting Numbers, Rev. ed. 1997, p. 169.

David A. Corneth, Table of n, a(n) for n = 1..10000 (first 1000 terms from Harry J. Smith)
Carlos Rivera, Puzzle 329. Odd abundant numbers not divided by 2 or 3, The Prime Puzzles and Problems Connection.
Jay L. Schiffman, Odd Abundant Numbers, Mathematical Spectrum, Volume 37, Number 2 (January 2005), pp 73-75.

Intersection of A005231 and A047201.
Cf. A006038, A047802, A110585.
Cf. A020338.

Select[ Range[ 1, 10^6, 2 ], DivisorSigma[ 1, # ] - 2# > 0 && Mod[ #, 5 ] != 0 & ]
ta={{0}};Do[g=n;s=DivisorSigma[1, n]-2*n; If[Greater[s, 0]&&!Equal[Mod[n, 2], 0]&& !Equal[Mod[n, 5], 0], Print[n];ta=Append[ta, n]], {n, 1, 2000000}] ta=Delete[ta, 1] (* Labos Elemer, Sep 15 2005 *)

{ n=0; forstep (m=1, 10^9, 2, if (m%5 && sigma(m) > 2*m, write("b064001.txt", n++, " ", m); if (n==1000, break)) ) } \\ Harry J. Smith, Sep 05 2009

More terms from Robert G. Wilson v, Sep 28 2001
Further terms from Labos Elemer, Sep 15 2005
Entry revised by N. J. A. Sloane, Mar 28 2006

A020331 Numbers whose base-3 representation is the juxtaposition of two identical strings.

Original entry on oeis.org

4, 8, 30, 40, 50, 60, 70, 80, 252, 280, 308, 336, 364, 392, 420, 448, 476, 504, 532, 560, 588, 616, 644, 672, 700, 728, 2214, 2296, 2378, 2460, 2542, 2624, 2706, 2788, 2870, 2952, 3034, 3116, 3198, 3280, 3362, 3444, 3526, 3608, 3690, 3772, 3854, 3936, 4018

Offset: 1

David W. Wilson, Melia Aldridge (ma38(AT)spruce.evansville.edu)

			50_10 = 1212_3. - _Jon E. Schoenfield_, Feb 11 2021

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A020330, A020332, A020333, A020334, A020335, A020336, A020337, A020338.

b3iQ[n_]:=Module[{idn3=IntegerDigits[n,3],len},len=Length[idn3];EvenQ[ len] && Take[idn3,len/2]==Take[idn3,-len/2 ]]; Select[Range[5000],b3iQ] (* Harvey P. Dale, Feb 08 2015 *)
a[n_] := n + n*3^Floor[Log[3, n] + 1]; Array[a, 50] (* Amiram Eldar, Apr 06 2021 *)

a(n) = n*3^floor(log_3(n)+1) + n. - Ilya Gutkovskiy, Jan 26 2018

cp's OEIS Frontend

A053061 a(n) is the decimal concatenation of n and n^2.

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A009470 a(n) is the concatenation of n and 8n.

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

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