A068142
a(0) = 21; for n > 0, a(n) is the smallest triangular number which is a (proper) multiple of a(n-1).
Original entry on oeis.org
21, 105, 210, 630, 25200, 32004000, 508031496000, 128015872500032496000, 3670698694547655407496988066168944000, 10302657959650317880463349610273001290502485245258650172717840000
Offset: 0
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.
-
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. *)
-
{a068142(m)=local(k,q,n); k=6; q=k*(k+1)/2; while(q0,k++; q=q+k))}
-
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
A068857
a(0) = 0, a(1) = 8; for n>=2: a(n) = smallest multiple of a(n-1) which is of the form 2k*(2k+2).
Original entry on oeis.org
0, 8, 24, 48, 288, 16128, 11950848, 4636929024, 88106288385024, 8038489644431643930624, 15177535939786079616000991061008232448, 40096515501441989312471498490435884509054125751527350190658560000
Offset: 0
24 = 4*6 is a member and the smallest multiple of 24 which is of the form 2k(2k+2) is 48 = 6*8.
-
m = 0; {0} ~Join~ Rest@ NestList[(m++; While[! Divisible[Set[k, # (# + 2) &[2 m]], #], m++]; k) &, 1, 8] (* Michael De Vlieger, Mar 18 2024 *)
-
from itertools import islice
from sympy import sqrt_mod_iter
def A068857_gen(): # generator of terms
yield 0
a = 8
while True:
yield a
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
A068857_list = list(islice(A068857_gen(),11)) # Chai Wah Wu, May 05 2024
A068513
a(0) = 15; for n > 0, a(n) is the smallest triangular number which is a (proper) multiple of a(n-1).
Original entry on oeis.org
15, 45, 630, 25200, 32004000, 508031496000, 128015872500032496000, 3670698694547655407496988066168944000, 10302657959650317880463349610273001290502485245258650172717840000
Offset: 0
-
Needs[ NumberTheory`NumberTheoryFunctions`]; 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] = 15; a[n_] := a[n] = nexttri[ a[n - 1]]; Table[ a[n], {n, 0, 8}]
-
from itertools import islice
from sympy import sqrt_mod_iter
def A068513_gen(): # generator of terms
a = 120
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
A068513_list = list(islice(A068513_gen(),10)) # Chai Wah Wu, May 05 2024
A093800
Triangular numbers such that the least common multiple of a pair of successive terms is triangular.
Original entry on oeis.org
1, 3, 6, 36, 2016, 26335, 53091360, 1420353154080, 70356589837262316000
Offset: 1
Showing 1-4 of 4 results.
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