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A353217 Triangular numbers (A000217) with arithmetic derivative (A003415) a palindrome (A002113).

%I A353217 #17 Jun 24 2022 17:20:20
%S A353217 0,1,3,6,10,15,136,153,231,741,1711,11026,22366,99681,104653,593505,
%T A353217 1348903,1378630,1886653,3098805,4388203,4474536,24587578,26626753,
%U A353217 32092066,45825951,132804253,165283471,197239591,355657785,498727153,866008153,1074091726,1144165366
%N A353217 Triangular numbers (A000217) with arithmetic derivative (A003415) a palindrome (A002113).
%e A353217 15 = A000217(5) and 15' = 8 = A002113(9), so 15 is a term.
%e A353217 153 = A000217(17) and 153' = 111 = A002113(21), so 153 is a term.
%t A353217 d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Table[n*(n + 1)/2, {n, 0, 50000}], PalindromeQ[d[#]] &] (* _Amiram Eldar_, Apr 30 2022 *)
%o A353217 (Magma) tr:=func<m|IsSquare(8*m+1)>; pal:=func<n|Intseq(n) eq Reverse(Intseq(n))>; f:=func<n |n le 1 select 0 else n*(&+[Factorisation(n)[i][2] / Factorisation(n)[i][1]: i in [1..#Factorisation(n)]])>; [n:n in [d*(d+1) div 2:d in [0..150000]]| pal(Floor(f(n)))];
%o A353217 (Python)
%o A353217 from itertools import chain, count, islice
%o A353217 from sympy import factorint
%o A353217 def A353217_gen(): # generator of terms
%o A353217     return chain((0,1),filter(lambda m:(s:=str(sum((m*e//p for p,e in factorint(m).items()))))[:(t:=(len(s)+1)//2)]==s[:-t-1:-1],(n*(n+1)//2 for n in count(2))))
%o A353217 A353217_list = list(islice(A353217_gen(),20)) # _Chai Wah Wu_, Jun 24 2022
%Y A353217 Cf. A000217, A003415, A002113, A068312.
%K A353217 nonn,base
%O A353217 1,3
%A A353217 _Marius A. Burtea_, Apr 30 2022