A119033 -id:A119033 - cp's OEIS frontend

A119034 Numbers k such that the k-th triangular number contains only digits {0,1,2}.

1, 4, 6, 15, 20, 66, 141, 200, 473, 666, 2000, 6666, 14840, 15620, 20000, 49193, 66666, 154926, 200000, 494006, 666666, 2000000, 4939858, 6666666, 20000000, 66666666, 200000000, 200105073, 651307933, 666666666, 1563400141

Offset: 1

Giovanni Resta, May 10 2006

Cross-references to similar sequences:
A119034 013 A119036 014 A119038 015 A119040 016 A119042 017 A119044
A119046 019 A119048 023 A119050 024 A218389 025 A119052 026 A119054
A218398 028 A119056 029 A119058 034 A119060 035 A119062 036 A119064
A119066 038 A119068 039 A119070 045 A119072 046 A119074 047 A218400
A119076 049 A119078 056 A119080 057 A119082 058 A119084 059 A119086
A119088 068 A119090 069 A119092 078 A119094 079 A218402 089 A119096
A119098 124 A119100 125 A119102 126 A119104 127 A119106 128 A119108
A119110 134 A119112 135 A119114 136 A119116 137 A119118 138 A119120
A119122 145 A119124 146 A119126 147 A119127 148 A119129 149 A119131
A119133 157 A119135 158 A119137 159 A119139 167 A119141 168 A119143
A119145 178 A119147 179 A119149 189 A119151 234 ... (*) 235 A119153
A119155 237 ... (*) 238 A119157 239 ... (*) 245 A119159 246 A119161
... (*) 248 A119163 249 ... (*) 256 A119165 257 A119167 258 A119169
A119171 267 A119173 268 A119175 269 A119177 278 A119179 279 ... (*)
A119181 345 A119183 346 A119185 347 ... (*) 348 ... (*) 349 ... (*)
A119187 357 A119189 358 A119191 359 A119193 367 A119195 368 A119197
A119199 378 A119201 379 ... (*) 389 ... (*) 456 A119203 457 A119205
A119207 459 A119209 467 A119211 468 A119213 469 A119215 478 A119217
... (*) 489 ... (*) 567 A119219 568 A119221 569 A119223 578 A119225
A119227 589 A119229 678 A119231 679 A119233 689 A119235 789 A119237
(*) Searched up to T(x)<10^32, resulted in short sequences (#terms<4):
247,249,279,479,489->{}, 234,237,239,347,348,349,379,389->{2}

G. Resta, Tridigital Triangular Numbers

Cf. A000217, A119033.

[n: n in [1..2*10^7] | Set(Intseq(Binomial(n+1, 2))) subset [0, 1, 2]]; // Vincenzo Librandi, Oct 07 2015

Select[Range[2 10^7], Complement[IntegerDigits[Binomial[# + 1, 2]], {0, 1, 2}] == {}&] (* Vincenzo Librandi, Oct 07 2015 *)

A118668 Number of distinct digits needed to write the n-th triangular number in decimal representation.

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 2, 3, 3, 3, 3, 3, 3, 3, 3, 3, 2, 3, 1, 3, 3, 3, 3, 3, 3, 3, 2, 4, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 4, 4, 2, 3, 4, 3, 4, 4, 3, 4, 2, 3, 4, 4, 4, 3, 3, 4, 3, 4, 3, 2, 4, 4, 4, 3, 3, 4, 4, 3, 4, 3, 4, 3, 4, 4, 4, 4, 3, 4, 3, 4, 4, 4, 2, 2, 3, 3, 4

Offset: 0

Reinhard Zumkeller, May 19 2006

0 < a(n) <= 10;

a(n) = A043537(A000217(n)).

			n=99: 99*(99+1)/2 = 4950 -> a(99) = #{0,4,5,9} = 4;
see A119033 for an overview of sequences with terms composed of not more than 3 distinct digits.
n=100: 100*(100+1)/2 = 5050 -> a(100) = #{0,5} = 2;

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000217, A043537, A045914, A213517, A213518, A255678.

a118668 = a043537 . a000217
a118668_list = map a043537 a000217_list
-- Reinhard Zumkeller, Jul 11 2015

Length[Union[IntegerDigits[#]]]&/@Accumulate[Range[0,110]] (* Harvey P. Dale, Jul 23 2012 *)

A213516 Triangular numbers having only 1 or 2 different digits in base 10.

Original entry on oeis.org

0, 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, 66, 78, 91, 171, 300, 595, 666, 990, 1711, 2211, 3003, 5050, 5151, 5565, 5995, 6555, 8778, 10011, 66066, 222111, 255255, 333336, 500500, 600060, 828828, 887778, 1188111, 5656566, 22221111, 50005000, 51151555, 88877778

Offset: 1

T. D. Noe, Jun 21 2012

The list of triangular numbers containing only one digit (A045914) is finite. This list is infinite because numbers like 8888777778, 222222111111, and 500000500000 occur an infinite number of times.

A309597 is a subsequence. - Seiichi Manyama, Sep 14 2019

Seiichi Manyama, Table of n, a(n) for n = 1..55

Cf. A000217, A045914, A213517, A309597, A319170.

Cf. A119033 (has list of sequences related to digits in triangular numbers).

[n*(n+1)/2: n in [0..10^5] | #Set(Intseq(n*(n+1) div 2)) le 2]; // Bruno Berselli, Oct 27 2012

t = {}; Do[tri = n*(n+1)/2; If[Length[Union[IntegerDigits[tri]]] <= 2, AppendTo[t, tri]], {n, 0, 10^5}]; t
Select[Accumulate[Range[0,20000]],Count[DigitCount[#],0]>7&] (* Harvey P. Dale, Sep 03 2020 *)

A119216 Triangular numbers composed of digits {4,7,8}.

Original entry on oeis.org

78, 8778, 448878, 887778, 7478778, 88877778, 8888777778, 44847878778, 747488474778, 888887777778, 88888877777778, 8888888777777778, 888888887777777778, 7778777484477877878, 88888888877777777778, 7444484774887874784778, 8888888888777777777778

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A053963, A119217. See A119033 for a table of cross-references.

Table[Select[FromDigits/@Tuples[{4,7,8},n],OddQ[Sqrt[8#+1]]&],{n,20}]//Flatten (* The program will take a long time to run. *) (* Harvey P. Dale, Oct 15 2017 *)

a(n) = A000217(A119217(n)). - Tyler Busby, Mar 26 2023

a(16)-a(17) from Tyler Busby, Mar 26 2023

A119230 Triangular numbers composed of digits {6,7,8}.

Original entry on oeis.org

6, 66, 78, 666, 6786, 8778, 676866, 887778, 88877778, 8888777778, 888887777778, 7768686676878, 68886888677778, 88888877777778, 8888888777777778, 888888887777777778, 88888888877777777778, 8888888888777777777778, 86686767888666878787778, 786777868668667866767766

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A119231. See A119033 for a table of cross-references.

a(n) = A000217(A119231(n)). - Tyler Busby, Mar 26 2023

a(19)-a(20) from Tyler Busby, Mar 26 2023

A119236 Triangular numbers composed of digits {7,8,9}.

Original entry on oeis.org

78, 8778, 887778, 88877778, 8888777778, 777887999778, 888887777778, 7798988788878, 88888877777778, 8888888777777778, 888888887777777778, 88888888877777777778, 8888888888777777777778, 78879897887889899897778, 888888888887777777777778, 88888888888877777777777778

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A058472, A119237. See A119033 for a table of cross-references.

a(n) = A000217(A119237(n)). - Michel Marcus, Mar 27 2023

a(15)-a(16) from Tyler Busby, Mar 27 2023

A079654 Triangular numbers using only the straight digits 1, 4 and 7.

Original entry on oeis.org

1, 171, 741, 1711, 117144471, 417417171, 7417744447141, 177141144447177414711441, 1417711714147441711474771771, 144114474147744777714441111111771711, 111411774474177744717747747477177774111

Offset: 1

Shyam Sunder Gupta, Jan 23 2003

Probably finite.

a(12) > 10^40 if it exists. - Tyler Busby, Mar 22 2023

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A028373, A053899, A119127. See A119033 for a table of cross-references.

Do[ If[ Union[ Join[ IntegerDigits[n(n + 1)/2], {1, 4, 7}]] == {1, 4, 7}, Print[n(n + 1)/2]], {n, 0, 3*10^7}]

a(n) = A000217(A119127(n)). - Tyler Busby, Mar 31 2023

Edited and extended by Robert G. Wilson v, Jan 24 2003
More terms from Giovanni Resta, May 10 2006
a(10)-a(11) from Tyler Busby, Mar 22 2023

A119035 Triangular numbers composed of digits {0,1,3}.

Original entry on oeis.org

1, 3, 10, 300, 3003, 10011, 303031, 1010331, 1030330, 1313010, 101111310, 113033130, 1033010331, 11031100311, 11033031331, 31031010003, 111111101310, 313313301003, 10113301131300, 3330130113300130003, 1113011003000100031311, 11013033300110103011310

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A119036. See A119033 for a table of cross-references.

[t: n in [1..2*10^7] | Set(Intseq(t)) subset {0,1,3} where t is n*(n+1) div 2]; // Vincenzo Librandi, Dec 18 2015

Rest[Select[FromDigits/@Tuples[{0, 1, 3}, 10], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* Vincenzo Librandi, Dec 18 2015 *)

a(n) = A000217(A119036(n)). - Tyler Busby, Mar 31 2023

a(21)-a(22) from Tyler Busby, Mar 22 2023

A119037 Triangular numbers composed of digits {0,1,4}.

Original entry on oeis.org

1, 10, 10011, 10440, 41041, 41001040, 141044410, 1010004040, 4104044101, 14041444410, 114011410040100, 440404440401100, 1041401040044401, 40114114410110140, 14001400401400441011, 1040410404040110411004140, 1014411414040144040404100100

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers

Cf. A000217, A058414, A119038. See A119033 for a table of cross-references.

[t: n in [1..2*10^7] | Set(Intseq(t)) subset {0,1,4} where t is n*(n+1) div 2]; // Vincenzo Librandi, Dec 18 2015

Rest[Select[FromDigits/@Tuples[{0, 1, 4}, 12], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* Vincenzo Librandi, Dec 18 2015 *)

a(n) = A000217(A119038(n)). - Tyler Busby, Mar 27 2023

a(17) from Tyler Busby, Mar 27 2023

A119039 Triangular numbers composed of digits {0,1,5}.

Original entry on oeis.org

1, 10, 15, 55, 105, 5050, 5151, 10011, 15051, 105111, 500500, 501501, 505515, 510555, 10015050, 50005000, 50015001, 50055015, 50105055, 51010050, 51111105, 51151555, 110551015, 1051501011, 1505550501, 5000050000, 5000150001, 5000550015, 5001050055, 5010055050

Offset: 1

Giovanni Resta, May 10 2006

Giovanni Resta, Tridigital Triangular Numbers.

Cf. A000217, A058416, A119040. See A119033 for a table of cross-references.

[t: n in [1..2*10^7] | Set(Intseq(t)) subset {0,1,5} where t is n*(n+1) div 2]; // Vincenzo Librandi, Dec 18 2015

Rest[Select[FromDigits/@Tuples[{0, 1, 5}, 12], IntegerQ[(Sqrt[8 # + 1] - 1)/2] &]] (* Vincenzo Librandi, Dec 18 2015 *)

a(n) = A000217(A119040(n)). - Tyler Busby, Mar 31 2023

More terms from Vincenzo Librandi, Dec 18 2015

cp's OEIS Frontend

A119034 Numbers k such that the k-th triangular number contains only digits {0,1,2}.

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A118668 Number of distinct digits needed to write the n-th triangular number in decimal representation.

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A213516 Triangular numbers having only 1 or 2 different digits in base 10.

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A119216 Triangular numbers composed of digits {4,7,8}.

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A119230 Triangular numbers composed of digits {6,7,8}.

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A119236 Triangular numbers composed of digits {7,8,9}.

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A079654 Triangular numbers using only the straight digits 1, 4 and 7.

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Mathematica

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A119035 Triangular numbers composed of digits {0,1,3}.

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Magma

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