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

1,3

Links

Table of n, a(n) for n=1..45.

Example

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.

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 *)

Prog

(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)))))];
(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

Crossrefs

Cf. A000217, A003415, A068346.

Sequence in context: A343811 A071299 A338767 * A061380 A350993 A308849

Adjacent sequences: A351128 A351129 A351130 * A351132 A351133 A351134

Keyword

nonn

Author

Marius A. Burtea, Feb 07 2022

Status

approved