A356748 Numbers k such that k and k+1 are both products of 2 triangular numbers.

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

Robert Israel, Table of n, a(n) for n = 1..138

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

Adjacent sequences: A356745 A356746 A356747 * A356749 A356750 A356751

Keyword

nonn

Author

Amiram Eldar, Aug 25 2022

Status

approved