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

0, 1, 3, 6, 10, 15, 136, 153, 231, 741, 1711, 11026, 22366, 99681, 104653, 593505, 1348903, 1378630, 1886653, 3098805, 4388203, 4474536, 24587578, 26626753, 32092066, 45825951, 132804253, 165283471, 197239591, 355657785, 498727153, 866008153, 1074091726, 1144165366

Offset

1,3

Links

Table of n, a(n) for n=1..34.

Example

15 = A000217(5) and 15' = 8 = A002113(9), so 15 is a term.
153 = A000217(17) and 153' = 111 = A002113(21), so 153 is a term.

Mathematica

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 *)

Prog

(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)))];
(Python)
from itertools import chain, count, islice
from sympy import factorint
def A353217_gen(): # generator of terms
    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))))
A353217_list = list(islice(A353217_gen(), 20)) # Chai Wah Wu, Jun 24 2022

Crossrefs

Cf. A000217, A003415, A002113, A068312.

Sequence in context: A130200 A202269 A151375 * A310082 A153453 A264041

Adjacent sequences: A353214 A353215 A353216 * A353218 A353219 A353220

Keyword

nonn,base

Author

Marius A. Burtea, Apr 30 2022

Status

approved