OFFSET
1,3
EXAMPLE
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