A187127 -id:A187127 - cp's OEIS frontend

A213518 Numbers k such that the triangular number k*(k+1)/2 has 2 different digits in base 10.

4, 5, 6, 7, 8, 9, 12, 13, 18, 24, 34, 44, 58, 66, 77, 100, 101, 105, 109, 114, 132, 141, 363, 666, 714, 816, 1000, 1095, 1287, 1332, 1541, 3363, 6666, 10000, 10114, 13332, 66666, 100000, 133332, 666666, 1000000, 1333332, 6666666, 10000000, 13333332, 33336636, 66666666, 100000000

Offset: 1

T. D. Noe, Jun 22 2012

The list of triangular numbers containing only one digit (A045914) is finite. This list is infinite because numbers like 133332, 666666, and 1000000 occur an infinite number of times.
A118668(a(n)) = 2. - Reinhard Zumkeller, Jul 11 2015
For n > 2, A325907(n) is a term. - Seiichi Manyama, Sep 15 2019

David Radcliffe, Table of n, a(n) for n = 1..116 (terms 1..51 from Seiichi Manyama, terms 52..60 from T. D. Noe).

Cf. A062691 (the corresponding triangular numbers), A213516, A213517, A325907.
Cf. A118668.
Cf. A187127.

a213518 n = a213518_list !! (n-1)
a213518_list = filter ((== 2) . a118668) [0..]
-- Reinhard Zumkeller, Jul 11 2015

t = {}; Do[tri = n*(n+1)/2; If[Length[Union[IntegerDigits[tri]]] == 2, AppendTo[t, n]], {n, 10^5}]; t

for(k=0, 1e8, if(#Set(digits(k*(k+1)/2))==2, print1(k", "))) \\ Seiichi Manyama, Sep 15 2019

a(45)-a(48) from Seiichi Manyama, Sep 15 2019

A227218 Smallest triangular number ending in n ones.

Original entry on oeis.org

1, 1711, 105111, 6271111, 664611111, 222222111111, 22222221111111, 2222222211111111, 2517912111111111, 18428299161111111111, 2222222222211111111111, 222222222222111111111111, 22222222222221111111111111, 1090161504430911111111111111

Offset: 1

Shyam Sunder Gupta, Sep 19 2013

If a triangular number ends in like digits then it can only end in 0's, 1's, 5's or 6's.

The sequence is infinite because the sequence of triangular numbers 21, 2211, 222111, 22221111, ... (A319170) is infinite.

			a(2) = 1711 because 1711 is the smallest triangular number ending in  2 '1's.

Chai Wah Wu, Table of n, a(n) for n = 1..501 (terms 1..200 from Giovanni Resta)

Cf. A000217, A187127, A227219, A227220, A319170.

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == (10^m - 1)/9, AppendTo[t, x]; Break[]], {n, 1, 10^m}], {m, 1, 10}]; t
a[n_] := Block[{x, y, s}, s = y /. List@ ToRules[ Reduce[(y+1)* y/2 == x*10^n +(10^n - 1)/9 && y > 0 && x >= 0, {y, x}, Integers] /. C[1] -> 0]; Min[s*(s + 1)/2]]; Array[a, 20] (* Giovanni Resta, Sep 20 2013 *)

from sympy import sqrt_mod_iter
def A227218(n):
    k, a = 10**n<<1, (10**n-1)//9<<1
    m = (a<<2)+1
    return min(b for b in ((d>>1)*((d>>1)+1) for d in sqrt_mod_iter(m, k) if d&1) if b%k==a)>>1 # Chai Wah Wu, May 04 2024

a(11)-a(14) from Giovanni Resta, Sep 20 2013

A227219 Smallest triangular number ending in at least n 5's.

Original entry on oeis.org

15, 55, 6555, 2235555, 717655555, 13140555555, 9206385555555, 1551745755555555, 103835341555555555, 26427628585555555555, 2262637355455555555555, 8808604932555555555555, 4704913988655555555555555, 4704913988655555555555555, 19391196055786555555555555555

Offset: 1

Shyam Sunder Gupta, Sep 19 2013

If a triangular number ends in like digits then it can only end in 0's, 1's, 5's or 6's.

			a(3)=6555 because 6555 is the smallest triangular number ending in 3 '5's.

Chai Wah Wu, Table of n, a(n) for n = 1..501

Cf. A000217, A187127, A227218, A227220.

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == 5*(10^m - 1)/9, AppendTo[t, x]; Break[]], {n, 1, 10^m}], {m, 1, 10}]; t
a[n_] := Block[{x, y, s}, s = y /. List@ ToRules[ Reduce[(y+1)* y/2 == x*10^n + 5*(10^n - 1)/9 && y > 0 && x >= 0, {y, x}, Integers] /. C[1] -> 0]; Min[s*(s + 1)/2]]; Array[a, 20] (* Giovanni Resta, Sep 20 2013 *)

from sympy import sqrt_mod_iter
def A227219(n):
    k, a = 10**n<<1, 10*(10**n-1)//9
    m = (a<<2)+1
    return min(b for b in ((d>>1)*((d>>1)+1) for d in sqrt_mod_iter(m, k) if d&1) if b%k==a)>>1 # Chai Wah Wu, May 04 2024

a(1) corrected and a(11)-a(15) from Giovanni Resta, Sep 20 2013

Name clarified by Michel Marcus, Sep 05 2019

A227220 Smallest triangular number ending in n 6's.

Original entry on oeis.org

6, 66, 666, 8146666, 2340066666, 109704666666, 15212926666666, 66488766666666, 173147535666666666, 11302559226666666666, 14642337066666666666, 77850029701666666666666, 1029964845546666666666666, 3086210323568466666666666666, 4014564887872666666666666666

Offset: 1

Shyam Sunder Gupta, Sep 19 2013

If a triangular number ends in like digits then it can only end in 0's, 1's, 5's or 6's.

			a(4)=8146666 because 8146666 is the smallest triangular number ending in  4 '6's.

Robert Israel, Table of n, a(n) for n = 1..500

Cf. A000217, A187127, A227218, A227219.

f:= proc(n) local k;
   k:=min(map(rhs@op, [msolve(k*(k+1)=4/3*(10^n-1),2*10^n)]));
   k*(k+1)/2
end proc:
map(f, [$1..30]); # Robert Israel, Sep 04 2019

t = {}; Do[Do[x = n*(n + 1)/2;If[Mod[x, 10^m] == 6*(10^m - 1)/9, AppendTo[t, x]; Break[]], {n, 1, 10^m}], {m, 1, 10}]; t
a[n_] := Block[{x,y,s}, s = y /. List@ ToRules[ Reduce[(y+1)* y/2 == x*10^n + 2(10^n - 1)/3 && y > 0 && x >= 0, {y, x}, Integers] /. C[1] -> 0]; Min[s*(s + 1)/2]]; Array[a,20] (* Giovanni Resta, Sep 20 2013 *)

from sympy import sqrt_mod_iter
def A227220(n):
    k, a = 10**n<<1, (10**n-1)//3<<2
    m = (a<<2)+1
    return min(b for b in ((d>>1)*((d>>1)+1) for d in sqrt_mod_iter(m, k) if d&1) if b%k==a)>>1 # Chai Wah Wu, May 04 2024

a(11)-a(15) from Giovanni Resta, Sep 20 2013

A088278 Smallest palindromic triangular numbers beginning with palindromes whose first digit is 1, 3, 5, 6, or 8.

Original entry on oeis.org

1, 3, 55, 6, 8778, 114401848104411, 0, 55, 66, 0, 10129457886113466431168875492101, 11121736463712111

Offset: 1

Amarnath Murthy, Sep 29 2003

The possible values of the ones digit of a triangular number are 0,1,3,5,6 and 8. Similarly, one can list the two-digit numbers k such that a triangular number of the form 100r + k can exist, and so on for the first three digits, etc. For palindromes P beginning with numbers other than these (e.g., for 33 and 88, which are two-digit palindromes P that start with 1, 3, 5, 6, or 8 but are not in A187127), the corresponding term is 0.

			From _Jon E. Schoenfield_, Mar 03 2018: (Start)
        Palindrome P       a(n) = smallest palindromic
       starting with    triangular number starting with P
   n  1, 3, 5, 6, or 8   (or 0 if no such number exists)
  ==  ================  =================================
   1          1                                         1
   2          3                                         3
   3          5                                        55
   4          6                                         6
   5          8                                      8778
   6         11                           114401848104411
   7         33                                         0
   8         55                                        55
   9         66                                        66
  10         88                                         0
  11        101          10129457886113466431168875492101
  12        111                         11121736463712111
  13        121                                         ?
  14        131                             1313207023131
  15        141                                         ?
  16        151                      15199896744769899151
  17        161                                         ?
  18        171                                       171
  19        181                                         ?
  20        191                                         ?
  21        303                                         0
(End)

Cf. A003098 (palindromic triangular numbers), A187127 (numbers that are the residue mod 100 of a triangular number). - Jon E. Schoenfield, Mar 03 2018

Name and Comments edited, offset changed to 1, and a(11) and a(12) corrected (a(11) taken from b-file at A003098) by Jon E. Schoenfield, Mar 03 2018

cp's OEIS Frontend

A213518 Numbers k such that the triangular number k*(k+1)/2 has 2 different digits in base 10.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A227218 Smallest triangular number ending in n ones.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A227219 Smallest triangular number ending in at least n 5's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

Extensions

A227220 Smallest triangular number ending in n 6's.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Python

Extensions

A088278 Smallest palindromic triangular numbers beginning with palindromes whose first digit is 1, 3, 5, 6, or 8.

Views

Author

Keywords

Comments

Examples

Crossrefs

Extensions