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A351131 Triangular numbers (A000217) whose second arithmetic derivative (A068346) is also a triangular number.

%I A351131 #14 Apr 01 2022 23:47:21
%S A351131 0,1,3,6,10,66,78,105,231,325,465,561,595,861,1378,2211,2278,2485,
%T A351131 3081,3570,3655,4278,4465,5253,6441,6670,8515,8778,9453,9870,10011,
%U A351131 10153,12561,13530,15051,18145,21115,21945,22578,23005,25878,27966,28441,40470,45753
%N A351131 Triangular numbers (A000217) whose second arithmetic derivative (A068346) is also a triangular number.
%e A351131 6 = A000217(3), 6'' = 5' = 1 = A000217(1), so 6 is a term.
%e A351131 66 = A000217(11), 66'' = 61' = 1 = A000217(1), so 66 is a term.
%e A351131 325 = A000217(25), 325'' = 155' = 36 = A000217(8), so 325 is a term.
%t A351131 d[0] = d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); Select[Table[n*(n + 1)/2, {n, 0, 300}], IntegerQ[Sqrt[8*d[d[#]] + 1]] &] (* _Amiram Eldar_, Feb 07 2022 *)
%o A351131 (Magma) tr:=func<m|IsSquare(8*m+1)>;  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..310]]| tr(Floor(f(Floor(f(n)))))];
%o A351131 (PARI) der(n) = my(f=factor(n)); vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
%o A351131 isok(m) = ispolygonal(m, 3) && ispolygonal(der(der(m)), 3); \\ _Michel Marcus_, Feb 16 2022
%o A351131 (Python)
%o A351131 from itertools import count, islice
%o A351131 from sympy import factorint, integer_nthroot, isprime, nextprime
%o A351131 def istri(n): return integer_nthroot(8*n+1, 2)[1]
%o A351131 def ad(n):
%o A351131     return 0 if n < 2 else sum(n*e//p for p, e in factorint(n).items())
%o A351131 def agen(): # generator of terms
%o A351131     for i in count(0):
%o A351131         t = i*(i+1)//2
%o A351131         if istri(ad(ad(t))):
%o A351131             yield t
%o A351131 print(list(islice(agen(), 45))) # _Michael S. Branicky_, Feb 16 2022
%Y A351131 Cf. A000217, A003415, A068346.
%K A351131 nonn
%O A351131 1,3
%A A351131 _Marius A. Burtea_, Feb 07 2022