A351130 Triangular numbers (A000217) whose arithmetic derivative (A003415) is also a triangular number.

0, 1, 3, 21, 351, 43956, 187578, 246753, 570846, 1200475, 4890628, 15671601, 83663580, 442903203, 3776109156, 35358717628, 1060996913571, 2443123072855, 65801068940503, 598914888327003, 1364298098094561

Offset

1,3

Comments

Triangular numbers in A229511.

Links

Table of n, a(n) for n=1..21.

Example

21 = A000217(6), 21' = 10 = A000217(4), so 21 is a term.
351 = A000217(26), 351' = 378 = A000217(27), so 351 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, 10^5}], IntegerQ[Sqrt[8*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..90000]]| tr(Floor(f(n)))];
(PARI) lista(nn) = my(t); for (n=0, nn, if (ispolygonal(der(t=n*(n+1)/2), 3), print1(t, ", "))); \\ 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(t)):
            yield t
print(list(islice(agen(), 14))) # Michael S. Branicky, Feb 16 2022

Crossrefs

Intersection of A000217 and A229511.

Cf. A003415.

Sequence in context: A083228 A376619 A052445 * A186271 A320949 A361056

Adjacent sequences: A351127 A351128 A351129 * A351131 A351132 A351133

Keyword

nonn,more

Author

Marius A. Burtea, Feb 07 2022

Extensions

a(20)-a(21) from Michael S. Branicky, Feb 17 2022

Status

approved