A394358 Integers k such that 8*k^2 + m*k is a pronic number (A002378) for some m in {-6, -2, 2, 6}.

0, 1, 2, 3, 5, 9, 20, 30, 55, 114, 174, 318, 667, 1015, 1856, 3885, 5915, 10815, 22646, 34476, 63037, 131988, 200940, 367404, 769285, 1171165, 2141390, 4483719, 6826049, 12480933, 26133032, 39785130, 72744211, 152314470, 231884730, 423984330, 887753791, 1351523251

Offset

1,3

Comments

The integer solutions exist when the discriminant 32*k^2 + 4*m*k + 1 is a perfect square.

Links

Table of n, a(n) for n=1..38.

Formula

8*k^2 + m*k = p*(p+1) for m in {-6, -2, 2, 6} and k, p >= 0.

Example

For k = 1, m = -6: 8*1^2 - 6*1 = 2, which is 1*2 (pronic).
For k = 2, m = -6: 8*2^2 - 6*2 = 20, which is 4*5 (pronic).
For k = 5, m = 2: 8*5^2 + 2*5 = 210, which is 14*15 (pronic).

Mathematica

oblongQ[n_] := IntegerQ @ Sqrt[4 n + 1]; q[k_]:=Max[Boole[oblongQ/@(8k^2+{-6, -2, 2, 6}k)]]==1; Select[Range[0, 10^5], q] (* James C. McMahon, Mar 26 2026 *)

Prog

(PARI)
is_pronic(x) = (x >= 0) && issquare(4*x + 1);
{ for(n=0, 1000,
    if(is_pronic(8*n^2 + 6*n) || is_pronic(8*n^2 - 6*n) || is_pronic(8*n^2 + 2*n) || is_pronic(8*n^2 - 2*n),
       print1(n, ", ")
    )
  )
}
(PARI) is_a394358(n) = forstep(k=-6, 6, 4, if(issquare(32*n^2 + 4*k*n + 1), return(1))); 0 \\ Hugo Pfoertner, Mar 17 2026

Crossrefs

Cf. A002378.

Sequence in context: A110542 A101542 A101581 * A365044 A392283 A349676

Adjacent sequences: A394355 A394356 A394357 * A394359 A394360 A394361

Keyword

nonn

Author

Mecibah Mohammed, Mar 17 2026

Extensions

More terms from Hugo Pfoertner, Mar 17 2026

Status

approved