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

Author

Amiram Eldar, Aug 25 2022

Keywords

• nonn

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

Examples

			9 is a term since 9 = 3*3 and 10 = 1*10 are both products of 2 triangular numbers.
		

Links

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

Crossrefs

Cf. A085780.

Programs

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 *)

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