A068127 -id:A068127 - cp's OEIS frontend

A068128 Triangular numbers with sum of digits = 6.

6, 15, 105, 231, 2211, 3003, 20301, 112101, 2003001, 122000010, 200030001, 20000300001, 2000003000001, 200000030000001, 20000000300000001, 2000000003000000001, 200000000030000000001, 20000000000300000000001, 2000000000003000000000001

Offset: 1

Amarnath Murthy, Feb 21 2002

1. The sequence is unbounded, as the (2*10^k +1)-th triangular number is a term. 2. The sum of the digits of triangular numbers in most cases is a triangular number. 3. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

Intersection of A000217 and A052220.

Cf. A068127.

Select[Accumulate[Range[20000002]],Total[IntegerDigits[#]]==6&] (* Harvey P. Dale, Jun 05 2012 *)

More terms from Sascha Kurz, Mar 06 2002
More terms from Sean A. Irvine, Mar 17 2023
Offset changed by Andrew Howroyd, Sep 17 2024

A068129 Triangular numbers with sum of digits = 10.

Original entry on oeis.org

28, 55, 91, 136, 190, 253, 325, 406, 703, 820, 1081, 1225, 1540, 1711, 2080, 2701, 3160, 3403, 5050, 7021, 10153, 11026, 12403, 15400, 17020, 20503, 21115, 23005, 24310, 32131, 41041, 51040, 52003, 60031, 72010, 80200, 90100, 106030, 110215

Offset: 1

Amarnath Murthy, Feb 21 2002

1. The sequence is unbounded, as the (2*10^k +2)-th triangular number is a term. 2. The sum of the digits of triangular numbers in most cases is a triangular number. 3. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

The second assertion above is wrong. Out of the first 100,000 triangular numbers, only 26,046 have a sum of their digits equal to a triangular number. - Harvey P. Dale, Jun 07 2017

Robert Israel, Table of n, a(n) for n = 1..164

Intersection of A000217 and A052224.

Cf. A062688, A068127, A068128.

for i from 1 to 9 do S[1,i]:= [i] od: S[1,10]:= []:
R:= NULL: count:= 0:
for d from 2 while count < 100 do
  for i from 1 to 10 do
    S[d,i]:= [seq(op(map(t -> 10*t + j, S[d-1,i-j])),j=0..i-1)];
  od:
  V:= select(t -> issqr(8*t+1), S[d,10]);
  if nops(V) > 0 then
    V:= sort(V);
    R:= R,op(V); count:= count+nops(V);
  fi
od:
R; # Robert Israel, May 15 2025

Select[Accumulate[Range[1000]],Total[IntegerDigits[#]]==10&] (* Harvey P. Dale, Jun 07 2017 *)

More terms from Sascha Kurz, Mar 06 2002

Offset changed by Andrew Howroyd, Sep 17 2024

A076713 Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits.

Original entry on oeis.org

1, 3, 6, 10, 21, 36, 45, 120, 153, 171, 190, 210, 300, 351, 378, 465, 630, 666, 780, 820, 990, 1035, 1128, 1275, 1431, 1540, 1596, 1770, 2016, 2080, 2556, 2628, 2850, 2926, 3160, 3240, 3321, 3486, 3570, 4005, 4465, 4560, 4950, 5050, 5460, 5565, 5778, 5886

Offset: 1

Shyam Sunder Gupta, Oct 26 2002

Intersection of A000217 and A005349. - K. D. Bajpai, Aug 13 2014

			a(5)=21: 21 is a triangular number and also a Harshad number as 21 is divisible by 2+1=3. So 21 is Harshad triangular number.

K. D. Bajpai, Table of n, a(n) for n = 1..15436
Shyam Sunder Gupta Fascinating Triangular Numbers

Cf. A000217, A005349. Includes A037156(n) for n >= 2. Includes A068127.

TriangularNumberQ[k_] := If[IntegerQ[1/2 (Sqrt[1 + 8 k] - 1)], True, False]; Harshad[k_] := Select[Range[k], IntegerQ[ #/(Plus @@ IntegerDigits[ # ])] &]; TriangularHarshad[k_] := Select[Harshad[k], TriangularNumberQ[#] &]; TriangularHarshad[5886] (* Ant King, Dec 13 2010 *)
A076713 = {}; Do[k = n*(n + 1)*1/2; If[IntegerQ[k/(Plus @@ IntegerDigits[k])], AppendTo[A076713, k]], {n,1000}]; A076713  (* K. D. Bajpai, Aug 13 2014 *)

A068130 Triangular numbers with sum of digits = 15.

Original entry on oeis.org

78, 276, 465, 528, 780, 861, 1176, 1275, 1653, 1770, 2346, 2850, 3570, 3741, 4371, 4560, 5253, 5460, 6216, 6441, 7260, 7503, 11175, 12246, 12561, 14028, 15225, 17205, 20706, 22155, 24090, 24531, 26106, 28203, 30381, 32640, 33153, 35511

Offset: 1

Amarnath Murthy, Feb 21 2002

1. The sequence is unbounded, as the (2*10^k +3)-th triangular number is a term. 2. The sum of the digits of triangular numbers in most cases is a triangular number. 3. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

Cf. A068127, A068128, A068129.

Select[Accumulate[Range[300]],Total[IntegerDigits[#]]==15&] (* Harvey P. Dale, Oct 09 2012 *)

More terms from Sascha Kurz, Mar 06 2002

A068131 Triangular numbers with sum of digits = 21.

Original entry on oeis.org

1596, 2775, 3486, 3828, 4278, 4656, 5565, 6555, 7626, 8256, 9453, 14196, 15753, 16653, 17391, 18336, 21945, 22791, 23871, 24753, 28920, 32385, 34716, 37128, 38226, 39621, 40755, 42195, 43365, 44850, 46056, 51681, 54615, 56280, 57630

Offset: 1

Amarnath Murthy, Feb 21 2002

1. The sequence is unbounded, as the (5*10^k + 6)-th triangular number is a term. 2. The sum of the digits of triangular numbers in most cases is a triangular number. 3. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Subsequence of A000217.

Cf. A068127, A068128, A068129, A068130.

Select[Accumulate[Range[400]],Total[IntegerDigits[#]]==21&] (* Harvey P. Dale, Jun 06 2015 *)

More terms from Sascha Kurz, Mar 06 2002

Offset changed by Andrew Howroyd, Sep 19 2024

A068133 First triangular number with digit sum = n-th triangular number.

Original entry on oeis.org

0, 1, 3, 6, 28, 78, 1596, 5995, 67896, 887778, 15997996, 398988876, 9876799878, 299789989975, 35998897988976, 589598998999878, 78999997699698778, 7987899888859999878, 1998997979958978979995, 539799799988999999688778

Offset: 0

Amarnath Murthy, Feb 21 2002

The sum of the digits of triangular numbers in most cases is a triangular number. Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.
From Jon E. Schoenfield, Jun 29 2010: (Start)
For any positive k < 132, it is true that more than half of the positive triangular numbers from T(1) through T(k) have a triangular digit sum. However, for any k > 132, more than half of the positive triangular numbers from T(1) through T(k) have a nontriangular digit sum. (At k = 132, there are 66 triangular and 66 nontriangular.)
There exist only finitely many triangular numbers whose digit sum is T(0)=0 or T(1)=1: T(0)=0 is, of course, the only one with digit sum 0, and T(1)=1 and T(4)=10 are the only two with digit sum 1. However, for digit sums equal to each of at least the next several triangular numbers, the conjecture can be easily confirmed by observing that, e.g., T(2), T(20), T(200), T(2000), etc., all have digit sum T(2)=3; T(2+1), T(20+1), T(200+1), T(2000+1), etc., all have digit sum T(3)=6; T(20+2), T(200+2), T(2000+2), T(20000+2), etc., all have digit sum T(4)=10; and, similarly, for all sufficiently large values of j, triangular numbers of the form T(2*10^j+m), where m = 3, 9, 23, 34, 132, 368, 1332, 3943, 19388, 88248, 244948, 1788848, 9838483, 19994343, respectively, will have digit sums T(5)=15, T(6)=21, ..., T(18)=171, respectively. (End)

Cf. A068127, A068128, A068129, A068130, A068131, A068132, A068134.

a(n) = A000217(A068134(n)). - Andrew Howroyd, Sep 21 2024

More terms from Larry Reeves (larryr(AT)acm.org), Jun 17 2002

Term a(0) inserted and terms a(18) and a(19) added by Jon E. Schoenfield, Jun 29 2010

A068132 Triangular numbers with sum of digits = 28.

Original entry on oeis.org

5995, 14878, 17578, 24976, 29890, 32896, 36856, 37675, 42778, 47278, 52975, 53956, 54946, 55945, 56953, 57970, 67528, 68635, 69751, 70876, 75466, 76636, 77815, 83845, 85078, 87571, 88831, 91378, 92665, 93961, 95266, 96580, 97903

Offset: 1

Amarnath Murthy, Feb 21 2002

The sequence is unbounded, as the (10^k + 9)-th triangular number for k>1 is a term.
The sum of the digits of triangular numbers in most cases is a triangular number.
Conjecture: For every triangular number T there exist infinitely many triangular numbers with sum of digits = T.

Subsequence of A000217.

Cf. A068127, A068128, A068129, A068130, A068131.

Select[Accumulate[Range[500]],Total[IntegerDigits[#]]==28&] (* Harvey P. Dale, Aug 30 2015 *)

More terms from Sascha Kurz, Mar 06 2002

Offset changed by Andrew Howroyd, Sep 19 2024

cp's OEIS Frontend

A068128 Triangular numbers with sum of digits = 6.

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A068129 Triangular numbers with sum of digits = 10.

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A076713 Harshad (Niven) triangular numbers: triangular numbers which are divisible by the sum of their digits.

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A068130 Triangular numbers with sum of digits = 15.

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A068131 Triangular numbers with sum of digits = 21.

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A068133 First triangular number with digit sum = n-th triangular number.

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A068132 Triangular numbers with sum of digits = 28.

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