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

0, 1, 3, 6, 10, 66, 78, 105, 231, 325, 465, 561, 595, 861, 1378, 2211, 2278, 2485, 3081, 3570, 3655, 4278, 4465, 5253, 6441, 6670, 8515, 8778, 9453, 9870, 10011, 10153, 12561, 13530, 15051, 18145, 21115, 21945, 22578, 23005, 25878, 27966, 28441, 40470, 45753

Offset: 1

Author

Marius A. Burtea, Feb 07 2022

Keywords

• nonn

Examples

			6 = A000217(3), 6'' = 5' = 1 = A000217(1), so 6 is a term.
66 = A000217(11), 66'' = 61' = 1 = A000217(1), so 66 is a term.
325 = A000217(25), 325'' = 155' = 36 = A000217(8), so 325 is a term.
		

Crossrefs

Cf. A000217, A003415, A068346.

Programs

Magma

tr:=func;  f:=func; [n:n in [d*(d+1) div 2:d in [0..310]]| tr(Floor(f(Floor(f(n)))))];

Mathematica

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 *)

PARI

der(n) = my(f=factor(n)); vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
isok(m) = ispolygonal(m, 3) && ispolygonal(der(der(m)), 3); \\ Michel Marcus, Feb 16 2022

Python

from itertools import count, islice
from sympy import factorint, integer_nthroot, isprime, nextprime
def istri(n): return integer_nthroot(8*n+1, 2)[1]
def ad(n):
    return 0 if n < 2 else sum(n*e//p for p, e in factorint(n).items())
def agen(): # generator of terms
    for i in count(0):
        t = i*(i+1)//2
        if istri(ad(ad(t))):
            yield t
print(list(islice(agen(), 45))) # Michael S. Branicky, Feb 16 2022