A058412 -id:A058412 - cp's OEIS frontend

A119033 Triangular numbers composed of digits {0,1,2}.

1, 10, 21, 120, 210, 2211, 10011, 20100, 112101, 222111, 2001000, 22221111, 110120220, 122000010, 200010000, 1210000221, 2222211111, 12001110201, 20000100000, 122021211021, 222222111111, 2000001000000, 12201101000011, 22222221111111, 200000010000000

Offset: 1

Giovanni Resta, May 10 2006

Cross-references to similar sequences:
A119033   013 A119035   014 A119037   015 A119039   016 A119041
A119043   018 A119045   019 A119047   023 A119049   024 A218390
A119051   026 A119053   027 A218397   028 A119055   029 A119057
A119059   035 A119061   036 A119063   037 A119065   038 A119067
A119069   045 A119071   046 A119073   047 A218399   048 A119075
A119077   056 A119079   057 A119081   058 A119083   059 A119085
A119087   068 A119089   069 A119091   078 A119093   079 A218401
A119095   123 A119097   124 A119099   125 A119101   126 A119103
A119105   128 A119107   129 A119109   134 A119111   135 A119113
A119115   137 A119117   138 A119119   139 A119121   145 A119123
A119125   147 A079654   148 A119128   149 A119130   156 A119132
A119134   158 A119136   159 A119138   167 A119140   168 A119142
A119144   178 A119146   179 A119148   189 A119150   234   {3}
A119152   236 A119154   237   {3}     238 A119156   239   {3}
A119158   246 A119160   247   { }     248 A119162   249   { }
A119164   257 A119166   258 A119168   259 A119170   267 A119172
A119174   269 A119176   278 A119178   279   { }     289 A119180
A119182   346 A119184   347   {3}     348   {3}     349   {3}
A119186   357 A119188   358 A119190   359 A119192   367 A119194
A119196   369 A119198   378 A119200   379   {3}     389   {3}
A119202   457 A119204   458 A119206   459 A119208   467 A119210
A119212   469 A119214   478 A119216   479   { }     489   { }
A119218   568 A119220   569 A119222   578 A119224   579 A119226
A119228   678 A119230   679 A119232   689 A119234   789 A119236
Entries marked "{ }" correspond to empty sequences: for every triangular number t, the residue t mod 100 contains at least one digit other than the three specified digits.
Entries marked "{3}" correspond to sequences containing only the single term 3: for every triangular number t != 3, the residue t mod 100 contains at least one digit other than the three specified digits.
(Proof: No triangular number ends in 2, 4, 7, or 9; every triangular number ending in 8 ends in 28 or 78; every triangular number ending in 3, other than the single-digit triangular number 3, ends in 03 or 53.) [Edited by Jon E. Schoenfield, May 02 2023]
Note that the first 36 sequences that are listed above do not contain "0" as the first term although 0 is a triangular number. In other words, sequences focus on the positive triangular numbers. - Altug Alkan, May 02 2016
a(n) == 1 or a(n) == 0 (mod 10). - Chai Wah Wu, Nov 30 2018

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A058412, A119034.

Cf. A213516 (triangular numbers having only two different digits).

[t: n in [1..2*10^7] | Set(Intseq(t)) subset {0,1,2} where t is n*(n+1) div 2]; // Vincenzo Librandi, Dec 18 2015

Rest[Select[FromDigits/@Tuples[{0, 1, 2}, 10], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* Vincenzo Librandi, Dec 18 2015 *)

isok(n) = ispolygonal(n, 3) && (vecmax(digits(n)) <= 2); \\ Michel Marcus, Dec 18 2015

a(n) = A000217(A119034(n)). - Tyler Busby, Mar 31 2023

a(24)-a(25) from Vincenzo Librandi, Dec 18 2015

A277959 Numbers k such that 2 is the largest decimal digit of k^2.

Original entry on oeis.org

11, 101, 110, 149, 1001, 1010, 1011, 1100, 1101, 1490, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 14499, 14900, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 144990, 149000, 316261

Offset: 1

Colin Barker, Nov 06 2016

The terms > 1 of A058411 can be considered as primitive elements of this sequence, obtained by multiplying those by powers of 10 (cf. formula). These terms of A058411 have at least 2 nonzero digits, and therefore their square has at least one digit 2. - M. F. Hasler, Nov 15 2017

M. F. Hasler, Table of n, a(n) for n = 1..1380 (first 50 terms from Colin Barker)
OEIS Index to sequences related to squares.

Cf. A277946 (the squares); A277960, A277961, A295005, ..., A295009 (analog for largest digit 3, 4, 5, ..., 9).
Cf. A058411, A058412 and A058413, ..., A058474. (Similar but no trailing 0's allowed.)
Cf. A136808 and A136809, ..., A137147 for other digit combinations. (Numbers must satisfy the same restriction as their squares.)

Select[Range[4*10^5], And[#[[2]] > 0, Union@ Take[RotateLeft[#, 2], 7] == {0}] &@ DigitCount[#^2] &] (* Michael De Vlieger, Nov 16 2017 *)

L=List(); for(n=1, 10000, if(vecmax(digits(n^2))==2, listput(L, n))); Vec(L)

A277959(LIM=1e15, L=List(), N=1)={while(LIM>N=next_A058411(N),my(t=N); until(LIMM. F. Hasler, Nov 15 2017

Equals (A058411 \ {1})*A011557, where A011557 = { 10^k; k >= 0 }. - M. F. Hasler, Nov 16 2017

Edited by M. F. Hasler, Nov 16 2017

A136808 Numbers k such that k and k^2 use only the digits 0, 1 and 2.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 1000000, 1000001, 1000010, 1000011, 1000100

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
Subsequence of A136809, A136816, ..., A136836. - M. F. Hasler, Jan 24 2008
A278038(18) = 10101, A136827(294) = 10110001101, A136831(1276) = 101100010001101 resp. A136836(1262) = 101090009991101 are the first terms from where on these four sequences differ from the present one. - M. F. Hasler, Nov 15 2017
From Jovan Radenkovicc, Nov 15 2024: (Start)
A nonnegative integer n is in this sequence iff 10*n is also in this sequence.
Not a subsequence of A278038 (binary numbers without '111'). A counterexample is 10^2884 + 10^2880 + 10^2872 + 10^2857 + 10^2497 + 10^2426 + 10^2285 + 10^2004 + 10^1443 + 10^1442 + 10^1441 + 10^881 + 10^600 + 10^459 + 10^388 + 10^27 + 10^12 + 10^4 + 1. There are infinitely many counterexamples not divisible by 10. This counterexample follows from the fact that 111^2+2000*4+200*4=12321+8000+800=21121. In fact, every binary substring will eventually occur in this sequence. Also, if n is a term containing only the digits 0 and 1, then 10^k*n+1 and n+10^k are also in this sequence for any sufficiently large integer k. (End)

			101000100100001^2 = 10201020220210222010200200001.

Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1359 terms from Jonathan Wellons)
Jonathan Wellons, Tables of Shared Digits [archived]
Index to sequences related to squares.

A subsequence of the binary numbers A007088.
Cf. A278038.
Cf. A136809, A136810, ..., A137147 for other digit combinations.
See also A058412 = A058411^2: squares having only digits {0,1,2}, A277946 = A277959^2 = squares whose largest digit is 2.

isA136808 := proc(n) local ndgs,n2dgs ; ndgs := convert(convert(n,base,10),set) ; n2dgs := convert(convert(n^2,base,10),set) ; if ( (ndgs union n2dgs) minus {0,1,2} ) = {} then true ; else false ; fi ; end: LtonRev := proc(L) local i ; add(op(i,L)*10^(i-1),i=1..nops(L)) ; end: A007089 := proc(n) convert(n,base,3) ; LtonRev(%) ; end: n := 1: for i from 0 do n3 := A007089(i) ; if isA136808(n3) then printf("%d %d ",n,n3) ; n := n+1 ; fi ; od: # R. J. Mathar, Jan 24 2008

Select[FromDigits/@Tuples[{0,1},7],Union[Take[DigitCount[#^2],{3,9}]]=={0}&] (* Harvey P. Dale, May 29 2013 *)

for(n=1,999,vecmax(digits((N=fromdigits(binary(n),10))^2))<3 && print1(N",")) \\ M. F. Hasler, Nov 15 2017

A058411 Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.

Original entry on oeis.org

1, 11, 101, 149, 1001, 1011, 1101, 10001, 10011, 11001, 14499, 100001, 100011, 100101, 101001, 110001, 316261, 1000001, 1000011, 1000101, 1010001, 1010011, 1100001, 1100101, 10000001, 10000011, 10000101, 10001001, 10001011, 10001101, 10010001, 10100001, 10100011, 10110001

Offset: 1

Patrick De Geest, Nov 15 2000

Sporadic solutions (not consisting only of digits 0 and 1): a(4) = 149, a(11) = 14499, a(17) = 316261, a(209) = 4604367505011, a(715) = 10959977245460011, a(1015) = 110000500908955011, a(1665) = 10099510939154979751, ... Three infinite subsequences are given by numbers of the form 10...01, 10...011 and 110...01, but there are many others. - M. F. Hasler, Nov 14 2017
From Zhao Hui Du, Mar 12 2024: (Start)
Most terms have a special pattern in that they have only digits 0 and 1 and could be written as Sum_{h=0..t} 10^x(h), where 2x(h) and x(h1)+x(h2) are distinct and x(0)=0 for the nonzero ending constraint. The number of n-digit terms in the sequence in the special pattern is A143823(n) - 2*A143823(n-1) + A143823(n-2) for n >= 2.
Terms with only digits 0 and 1 but not in the special pattern exist as well. If we define f(x) = 1 + x^768 + x^960 + x^1008 + x^1020 + x^1028 + x^1040 + x^1088 + x^1280 + x^2048, f(x)^2 is a function with all nonzero coefficients 1,2,10 (the only coefficient of x^2048 is 10 and the coefficient of x^2049 is 0). So f(10) is in the sequence but not in the special pattern. (End)

Zhao Hui Du, Table of n, a(n) for n = 1..4000 (first 1000 terms from Chai Wah Wu; 1001..1269 from M. F. Hasler)
Patrick De Geest, Index to related sequences.
Hisanori Mishima, Sporadic tridigital solutions.
OEIS Wiki, Index to sequences related to squares having only given digits.

Cf. A058412 (the squares); A058412, ..., A058474 (other 3-digit combinations).

Cf. A063009, A066139. - Zak Seidov, Jul 01 2013

[n: n in [1..2*10^8 by 2] | Set(Intseq(n^2)) subset [0,1,2]]; // Vincenzo Librandi, Feb 24 2016

R[1]:= {1,9};
for m from 2 to 10 do
  R[m]:= select(t -> max(convert(t^2 mod 10^m, base, 10)) <= 2, map(s -> seq(s + i*10^(m-1),i=0..9), R[m-1]))
od:
Res:= {seq(op(select(t -> t >= 10^(m-1) and max(convert(t^2,base,10)) <= 2, R[m])),m=1..10)}:
sort(convert(Res,list)); # Robert Israel, Feb 23 2016

Select[Range[10^6], And[Total@ Take[RotateRight@ DigitCount@ #, -7] == 0, Mod[#, 10] != 0] &[#^2] &] (* Michael De Vlieger, Nov 14 2017 *)

isok(n)={ n%10 && vecmax(digits(n^2)) < 3 } \\ Michel Marcus, Feb 24 2016, edited by M. F. Hasler, Nov 14 2017

A058411_list = [i for i in range(10**6) if i % 10 and max(str(i**2)) < '3'] # Chai Wah Wu, Feb 23 2016

a(n) = sqrt(A058412(n)). - Zak Seidov, Jul 01 2013

b-file corrected by Zhao Hui Du, Mar 07 2024

A277948 Squares whose largest decimal digit is 4.

Original entry on oeis.org

4, 144, 324, 400, 441, 1024, 1444, 2304, 2401, 10404, 14400, 23104, 32041, 32400, 33124, 40000, 40401, 44100, 101124, 102400, 103041, 110224, 114244, 121104, 131044, 144400, 203401, 204304, 213444, 230400, 232324, 240100, 300304, 301401, 421201, 1004004

Offset: 1

Colin Barker, Nov 05 2016

A subsequence of A158082, in turn a subsequence of A000290.

Colin Barker, Table of n, a(n) for n = 1..1000

Cf. A000290 (the squares).
Cf. A277961 (square roots of these terms).
Cf. A277946, A277947, A295015, ..., A295019 (analog for largest digit = 2, 3, 5, ..., 9).
Cf. A058412, A058411, ..., A058474 and A136808, A136809, ..., A137147 for other restrictions on digits of squares.

[n^2: n in [1..1000000] | Maximum(Intseq(n^2)) eq 4]; // Vincenzo Librandi, Nov 06 2016

Select[Range[1100]^2,Max[IntegerDigits[#]]==4&] (* Harvey P. Dale, Jul 01 2017 *)

L=List(); for(n=1, 10000, if(vecmax(digits(n^2))==4, listput(L, n^2))); Vec(L)

a(n) = A277961(n)^2. - M. F. Hasler, Nov 12 2017

Intersection of A000290 and A277966. - M. F. Hasler, Nov 15 2017

A030175 When squared gives number composed of digits {1,2,3}.

Original entry on oeis.org

1, 11, 111, 36361, 363639, 461761, 3636361, 34815389, 362397739, 176412364139, 57637950363639, 3497458093147239, 56843832676142723489, 557963558954625926861

Offset: 1

Patrick De Geest

P. De Geest, Squares containing at most three distinct digits, Index entries for related sequences
Author?, Source(txt)
Hisanori Mishima, Sporadic tridigital solutions
Eric Weisstein's World of Mathematics, Square Number.
Index entries for sequences related to squares.

Cf. A030174; A000290.
Cf. A006716 = A027675^2, A030176 = A030177^2, A058411^2 = A058412, ..., A058473^2 = A058474.
Cf. A136808, A136809, ..., A137147: n and n^2 have digits {...}.
Cf. A277959^2 = A277946 and A277960^2 = A277947: squares whose largest digit is 2 resp. 3.

Do[ If[ Union[ Join[{1, 2, 3}, IntegerDigits[n^2] ] ] == {1, 2, 3}, Print[n] ], {n, 0, 10^9}]

lista(nn) = for(n=1, nn, if(setminus(vecsort(digits(n^2), , 8), [1, 2, 3])==[], print1(n, ", "))) \\ Iain Fox, Nov 16 2017

a(n)^2 = A030174(n). - M. F. Hasler, Nov 16 2017

More terms from Patrick De Geest, Mar 01 2000
More terms from C. Ronaldo (aga_new_ac(AT)hotmail.com), Jan 14 2005
Offset corrected by Iain Fox, Nov 16 2017

A136836 Numbers k such that k and k^2 use only the digits 0, 1, 2 and 9.

Original entry on oeis.org

0, 1, 10, 11, 100, 101, 110, 1000, 1001, 1010, 1011, 1100, 1101, 10000, 10001, 10010, 10011, 10100, 10110, 11000, 11001, 11010, 100000, 100001, 100010, 100011, 100100, 100101, 100110, 101000, 101001, 101100, 110000, 110001, 110010, 110100, 1000000, 1000001, 1000010, 1000011, 1000100

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.
Comparison of b-files indicates that the first difference from A136831 is at the 1262nd entry. - R. J. Mathar Apr 29 2008
More precisely, A278038(18) = 10101, A136827(294) = 10110001101 resp. A136808(1262) = A136831(1262) = 101100000000000 are the first terms from where on these four sequences differ from the present one; a(1262) = 101090009991101 is also the first term containing a digit > 1. - M. F. Hasler, Nov 15 2017

			101090009991101^2 = 10219190120000900002099192201.

Jonathan Wellons, Table of n, a(n) for n = 1..1360
J. Wellons, Tables of Shared Digits [archived]

Cf. A136808, A136809, A136810, ..., A137147 for other digit combinations.
See also A058412 = A058411^2: squares having only digits {0,1,2}, A277946 = A277959^2 = squares whose largest digit is 2.
The first 1261 terms are also a subsequence of A278038 (binary numbers without '111'), in turn a subsequence of the binary numbers A007088.

With[{c={0,1,2,9}},Select[FromDigits/@Tuples[c,7],SubsetQ[c,IntegerDigits[#^2]]&]] (* Harvey P. Dale, Feb 11 2024 *)

A136812 Numbers k such that k and k^2 use only the digits 0, 1, 2, 3 and 6.

Original entry on oeis.org

0, 1, 6, 10, 11, 60, 100, 101, 106, 110, 111, 361, 600, 601, 1000, 1001, 1006, 1010, 1011, 1060, 1100, 1101, 1106, 1110, 1631, 3606, 3610, 6000, 6001, 6010, 6011, 10000, 10001, 10006, 10010, 10011, 10060, 10100, 10101, 10106, 10110, 10111, 10301, 10306, 10600, 11000, 11001, 11006, 11010, 11060, 11100, 11101, 16310, 32111, 36060, 36100, 36361

Offset: 1

Jonathan Wellons (wellons(AT)gmail.com), Jan 22 2008

Generated with DrScheme.

			1031316261^2 = 1063613230203020121.

Jonathan Wellons, Table of n, a(n) for n = 1..1315
J. Wellons, Tables of Shared Digits [archived]
Index entries for sequences related to squares.

Cf. A136808, ..., A137147.

Cf. A006716 = A027675^2, A030174 = A030175^2, A030176 = A030177^2, A058411^2 = A058412, ..., A058473^2 = A058474.

cp's OEIS Frontend

A119033 Triangular numbers composed of digits {0,1,2}.

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A277959 Numbers k such that 2 is the largest decimal digit of k^2.

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A136808 Numbers k such that k and k^2 use only the digits 0, 1 and 2.

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A058411 Numbers k such that k^2 contains only digits {0,1,2}, not ending with zero.

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A277948 Squares whose largest decimal digit is 4.

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A030175 When squared gives number composed of digits {1,2,3}.

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A136836 Numbers k such that k and k^2 use only the digits 0, 1, 2 and 9.

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