A394731 Numbers k such that k*(k + 1)/2 + d is a triangular number (A000217), where d > (k + 1) is a divisor of k*(k + 1)/2.

14, 44, 84, 90, 152, 230, 324, 434, 440, 492, 560, 702, 860, 1034, 1224, 1260, 1430, 1652, 1890, 2144, 2414, 2700, 2736, 2870, 3002, 3320, 3654, 4004, 4360, 4370, 4752, 5060, 5150, 5564, 5994, 6440, 6902, 7380, 7874, 8384, 8424, 8910, 9452, 10010, 10584

Offset

1,1

Comments

k = 2*A033954(n > 0) and k = A053141(n > 1), but it does not cover all terms.

For integer x > 0, k = 2*x*S(n, 4*x + 2) where S(n, 4*x + 2) denotes the partial sums of the Chebyshev sequence.

Links

Table of n, a(n) for n=1..45.

Example

For k = 14: 14*15/2 + 105 = 210 is a triangular number, thus 14 is a term.

Mathematica

q[k_] := AnyTrue[k*(k+1)/2 + Select[Divisors[k*(k+1)/2], # > k+1 &], IntegerQ[Sqrt[8*#+1]] &]; Select[Range[11000], q] (* Amiram Eldar, Mar 30 2026 *)

Prog

(PARI) isok(k) = my(t=k*(k+1)/2); fordiv(t, d, if ((d>k+1) && ispolygonal(t+d, 3), return(1))); \\ Michel Marcus, Apr 11 2026
(Python)
from itertools import count, islice
from sympy import divisors
from sympy.ntheory.primetest import is_square
def A394731_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda k:any(is_square((k*(k+1)+(d<<1)<<2)|1) for d in divisors(k*(k+1)>>1, generator=True) if d>k+1), count(max(startvalue, 1)))
A394731_list = list(islice(A394731_gen(), 45)) # Chai Wah Wu, Apr 19 2026

Crossrefs

Cf. A000217, A033954, A053141, A075528.

Sequence in context: A064125 A089031 A387733 * A265152 A033587 A189807

Adjacent sequences: A394728 A394729 A394730 * A394732 A394733 A394734

Keyword

nonn

Author

Ctibor O. Zizka, Mar 30 2026

Status

approved