A394731 - OEIS

%I #30 Apr 20 2026 00:39:53

%S 14,44,84,90,152,230,324,434,440,492,560,702,860,1034,1224,1260,1430,

%T 1652,1890,2144,2414,2700,2736,2870,3002,3320,3654,4004,4360,4370,

%U 4752,5060,5150,5564,5994,6440,6902,7380,7874,8384,8424,8910,9452,10010,10584

%N 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.

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

%C 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.

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

%t 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 *)

%o (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

%o (Python)

%o from itertools import count, islice

%o from sympy import divisors

%o from sympy.ntheory.primetest import is_square

%o def A394731_gen(startvalue=1): # generator of terms >= startvalue

%o     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)))

%o A394731_list = list(islice(A394731_gen(),45)) # _Chai Wah Wu_, Apr 19 2026

%Y Cf. A000217, A033954, A053141, A075528.

%K nonn

%O 1,1

%A _Ctibor O. Zizka_, Mar 30 2026