A076713 -id:A076713 - cp's OEIS frontend

A242043 Pentagonal numbers that are also Niven numbers.

1, 5, 12, 70, 117, 210, 247, 330, 715, 782, 1080, 1520, 1926, 2625, 2752, 3015, 3290, 3432, 4510, 5370, 5922, 6902, 7740, 8400, 9560, 11310, 12105, 13776, 14652, 15862, 17442, 21182, 21540, 23002, 24130, 26070, 27270, 30602, 31032, 32340, 34580, 38320, 39285

Offset: 1

K. D. Bajpai, Aug 12 2014

Intersection of A000326 and A005349.

			a(3) = 12 = 3*(3 * 3 - 1)/2 is a pentagonal number. Since 12 is divisible by 1 + 2 = 3, it is also a Harshad number and therefore in the sequence.
a(5) = 117 = 9*(3 * 9 - 1)/2 is a pentagonal number. Since 117 is divisible by 1 + 1 + 7 = 9 is also a Harshad number, and therefore in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..10344

Cf. A000326, A005349, A076713.

A242043 = {}; Do[k = (3*n^2 - n)/2; If[IntegerQ[k/(Plus @@ IntegerDigits[k])], AppendTo[A242043, k]], {n, 300}]; A242043 (* Bajpai *)
Select[Table[n(3n - 1)/2, {n, 200}], Divisible[#, Plus@@IntegerDigits[#]] &] (* Alonso del Arte, Aug 16 2014 *)
Select[PolygonalNumber[5,Range[200]],Mod[#,Total[IntegerDigits[#]]]==0&] (* Harvey P. Dale, Nov 30 2022 *)

A243008 Triangular numbers divisible by the square of the sum of their digits.

Original entry on oeis.org

1, 10, 3240, 3321, 13041, 13203, 15400, 65341, 80200, 90100, 161028, 210276, 260281, 265356, 266085, 300700, 346528, 500500, 937765, 947376, 1043290, 1228528, 1313010, 1628110, 2049300, 2390391, 2421100, 3357936, 3746953, 4020030, 5250420, 6641190, 6857956, 6939675

Offset: 1

K. D. Bajpai, Aug 20 2014

Intersection of A000217 and A072081.

			a(3) = 3240 = 80 * (80 + 1)/2 is a triangular number. Since 3240 is divisible by (3 + 2 + 4 + 0)^2 = 81, it appears in the sequence.
a(3) = 3321 = 81 * (81 + 1)/2 is a triangular number. Since 3321 is divisible by (3 + 3 + 2 + 1)^2 = 81, it appears in the sequence.

K. D. Bajpai, Table of n, a(n) for n = 1..12445

Cf. A000217, A072081, A076713, A118693.

Select[Table[n*(n + 1)/2, {n, 10000}], Divisible[#, Plus @@ IntegerDigits[#]^2] &]

for(n=1,10^4,s=n*(n+1)/2;if(s%(sumdigits(s)^2)==0,print1(s,", "))) \\ Derek Orr, Aug 23 2014

A375824 Triangular numbers whose sum of digits is 9.

Original entry on oeis.org

36, 45, 153, 171, 351, 630, 1035, 1431, 2016, 3240, 3321, 4005, 8001, 10440, 13041, 13203, 16110, 21321, 23220, 25200, 101025, 105111, 114003, 222111, 320400, 321201, 1010331, 1241100, 1313010, 1400301, 2013021, 2031120, 2410110, 4020030, 10006101, 11203011, 20012301, 32004000, 32012001, 33020001

Offset: 1

Robert Israel, Aug 30 2024

Infinite subsequences include 2 * 10^(2*k) + 13 * 10^k + 21, 2 * 10^(2*k) + 31 * 10^k + 120, 32 * 10^(2*k) + 4 * 10^k, and 32 * 10^(2*k) + 12 * 10^k + 1.

Conjecture: the last term not of one of those subsequences is a(53) = 210010000005.

			a(4) = 153 is a term because 153 = 17 * 18/2 is a triangular number and 1 + 5 + 3 = 9.

Robert Israel, Table of n, a(n) for n = 1..95

Intersection of A000217 and A052223. Contained in A117404 and A076713.

F:= proc(d,s) option remember;
# d-digit numbers with sum of digits s
      local R,i;
      R:= {};
      for i from 0 to min(s,9) do
        R:= R union map(t -> 10*t+i, procname(d-1,s-i))
      od;
      R
end proc:
F(1,0):= {}:
for i from 1 to 9 do F(1,i):= {i} od:
sort(convert(`union`(seq(select(t -> issqr(1+8*t), F(d,9)),d=1..12)),list));

Select[Range[10000](Range[10000]+1)/2,DigitSum[#]==9 &] (* Stefano Spezia, Sep 01 2024 *)

select(x->(sumdigits(x)==9), vector(10000, n, n*(n+1)/2)) \\ Michel Marcus, Aug 31 2024

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A242043 Pentagonal numbers that are also Niven numbers.

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A243008 Triangular numbers divisible by the square of the sum of their digits.

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A375824 Triangular numbers whose sum of digits is 9.

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