A387123 Numbers k such that Sum_{i=1..r} (k-i) and Sum_{i=1..r} (k+i) are both triangular for some r with 1 <= r < k.

2, 6, 9, 21, 24, 38, 50, 53, 65, 77, 90, 96, 104, 133, 147, 195, 201, 224, 247, 286, 324, 377, 450, 483, 553, 588, 605, 614, 713, 792, 901, 1014, 1029, 1043, 1066, 1074, 1155, 1274, 1349, 1575, 1784, 1885, 1920, 2034, 2057, 2109, 2279, 2312, 2342, 2622

Offset: 1

Author

Ctibor O. Zizka, Aug 17 2025

Keywords

• nonn
• new

Comments

For m >= 1, if k = m*(m+1)^2/2 then r = m, thus A006002 is a subsequence. For k >= 286 from A101265 or A101879, r = k-1.

Examples

			For k = 6: the least r = 5, T_i = 1 + 2 + 3 + 4 + 5 = 15, T_j = 7 + 8 + 9 + 10 + 11 = 45, both T_i and T_j are triangular numbers, thus k = 6 is a term.
		

Crossrefs

Cf. A000217, A006002, A101265, A101879.

Programs

Mathematica

triQ[n_] := IntegerQ[Sqrt[8*n + 1]]; q[k_] := Module[{r = 1, s1 = 0, s2 = 0}, While[s1 += k - r; s2 += k + r; r < k && (! triQ[s1] || ! triQ[s2]), r++]; 1 <= r < k]; Select[Range[3000], q] (* Amiram Eldar, Aug 17 2025 *)

PARI

isok(k) = my(sm=0, sp=0); for (r=1, k-1, sm+=k-r; sp+=k+r; if (ispolygonal(sm, 3) && ispolygonal(sp, 3), return(r));); \\ Michel Marcus, Aug 17 2025

Python

from itertools import count, islice
from sympy.ntheory.primetest import is_square
def A387123_gen(startvalue=1): # generator of terms >= startvalue
    for k in count(max(startvalue,1)):
        if any(is_square(((k*r<<1)-r*(r+1)<<2)+1) and is_square(((k*r<<1)+r*(r+1)<<2)+1) for r in range(1,k)):
            yield k
A387123_list = list(islice(A387123_gen(),50)) # Chai Wah Wu, Aug 21 2025