login
A351131
Triangular numbers (A000217) whose second arithmetic derivative (A068346) is also a triangular number.
0
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
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
KEYWORD
nonn
AUTHOR
Marius A. Burtea, Feb 07 2022
STATUS
approved