A068142 a(0) = 21; for n > 0, a(n) is the smallest triangular number which is a (proper) multiple of a(n-1).

21, 105, 210, 630, 25200, 32004000, 508031496000, 128015872500032496000, 3670698694547655407496988066168944000, 10302657959650317880463349610273001290502485245258650172717840000

Offset: 0

Author

Amarnath Murthy, Feb 22 2002

Keywords

• nonn

Examples

			a(1) = 105, since 105 = 5*21 = 5*a(0), 105 is a triangular number and 2*a(0) = 42, 3*a(0) = 63, 4*a(0) = 84 are not triangular numbers.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..11

Crossrefs

Cf. A000217, A068776.

Programs

Mathematica

pm1[{k_}] := {1, k-1}; pm1[lst_] := Module[{a, m, v}, a=lst[[1]]; m=Times@@Rest[lst]; v=pm1[Rest[lst]]; Union[ChineseRemainder[{1, #}, {a, m}]&/@v, ChineseRemainder[{-1, #}, {a, m}]&/@v]]; nexttri[1]=3; nexttri[n_] := Module[{s}, s=(pm1[Power@@#&/@FactorInteger[4n]]^2-1)/8; For[i=1, True, i++, If[s[[i]]>n, Return[s[[i]]]]]]; a[0]=21; a[n_] := a[n]=nexttri[a[n-1]]; (* First do <= 3, pm1[lst] is the list of numbers less than their product and == 1 or -1 (mod every element of lst). nexttri[n] is the smallest triangular number >n and divisible by n. *)

PARI

{a068142(m)=local(k,q,n); k=6; q=k*(k+1)/2; while(q0,k++; q=q+k))}

Python

from itertools import islice
from sympy import sqrt_mod_iter
def A068142_gen(): # generator of terms
    a = 168
    while True:
        yield a>>3
        b = a+1
        for d in sqrt_mod_iter(1,a):
            if d==1 or d**2-1 == a:
                d += a
            if d&1 and d < b:
                b = d
        a = b**2-1
A068142_list = list(islice(A068142_gen(),12)) # Chai Wah Wu, May 05 2024

Extensions

Edited and extended by Klaus Brockhaus, Robert G. Wilson v, Mar 01 2002 and Dean Hickerson, Mar 09 2002