login
A356748
Numbers k such that k and k+1 are both products of 2 triangular numbers.
1
0, 9, 90, 135, 945, 1710, 1890, 4959, 5670, 8910, 10584, 11025, 11934, 13860, 19305, 21735, 26334, 32130, 36855, 44550, 49140, 65340, 107415, 138600, 172080, 239085, 305370, 351540, 366795, 459360, 849555, 873180, 933660, 1100385, 1413720, 1516410, 1904175, 2297295
OFFSET
1,2
COMMENTS
Numbers k such that k and k+1 are both terms of A085780.
Are all the terms divisible by 9?
Yes, because the product of two triangular numbers == 0, 1, 3 or 6 (mod 9). - Robert Israel, Apr 05 2023
LINKS
EXAMPLE
9 is a term since 9 = 3*3 and 10 = 1*10 are both products of 2 triangular numbers.
MAPLE
N:= 10^9: # for terms <= N
S:= {0}:
for x from 1 do
s:= x*(x+1)/2;
if s^2 > N then break fi;
for y from x do
t:= y*(y+1)/2;
if s*t > N then break fi;
S:= S union {s*t};
od od:
L:= sort(convert(S, list)):
DL:= L[2..-1]-L[1..-2]:
J:= select(t -> DL[t]=1, [$1..nops(DL)]):
L[J]; # Robert Israel, Apr 05 2023
MATHEMATICA
t = Table[n*(n + 1)/2, {n, 0, 3000}]; s = Select[Union[Flatten[Outer[Times, t, t]]], # <= t[[-1]] &]; i = Position[Differences[s], 1] // Flatten; s[[i]]
Take[Select[Partition[Union[Times@@@Tuples[Accumulate[Range[0, 2500]], 2]], 2, 1], #[[2]] - #[[1]]==1&][[All, 1]], 40] (* Harvey P. Dale, Oct 23 2022 *)
PROG
(Python)
from itertools import count, islice
from sympy import divisors, integer_nthroot
def A356748_gen(startvalue=0): # generator of terms >= startvalue
if startvalue <= 0:
yield 0
flag = False
for n in count(max(startvalue, 1)):
for d in divisors(m:=n<<2):
if d**2 > m:
flag = False
break
if integer_nthroot((d<<2)+1, 2)[1] and integer_nthroot((m//d<<2)+1, 2)[1]:
if flag: yield n-1
flag = True
break
else:
flag = False
A356748_list = list(islice(A356748_gen(), 10)) # Chai Wah Wu, Aug 28 2022
CROSSREFS
Cf. A085780.
Sequence in context: A061478 A061878 A341010 * A140160 A043960 A044641
KEYWORD
nonn
AUTHOR
Amiram Eldar, Aug 25 2022
STATUS
approved