A309355 Even numbers k such that k! is divisible by k*(k+1)/2.

8, 14, 20, 24, 26, 32, 34, 38, 44, 48, 50, 54, 56, 62, 64, 68, 74, 76, 80, 84, 86, 90, 92, 94, 98, 104, 110, 114, 116, 118, 120, 122, 124, 128, 132, 134, 140, 142, 144, 146, 152, 154, 158, 160, 164, 168, 170, 174, 176, 182, 184, 186, 188, 194, 200, 202, 204, 206

Offset: 1

Author

Gerhard Palme, Jul 25 2019

Keywords

• nonn

Comments

Even terms in A060462.

And A071904 are the successors of a(n).

Even numbers that are not a prime - 1. That is, even numbers not in A006093. - Terry D. Grant, Oct 31 2020

Examples

			8! = 40320 is divisible by 8*9/2 = 36.
14! is divisible by 14*15/2.
		

References

• J. D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 98, pp. 29; 145-146, MAA Washington DC, 1996.
• Die WURZEL - Zeitschrift für Mathematik, 53. Jahrgang, Juli 2019, S. 171, WURZEL-Aufgabe 2019-36 von Gerhard Dietel, Regensburg.

Crossrefs

Cf. A060462, A071904.

Essentially the same as A186193.

Cf. A006093.

Programs

Magma

[k: k in [2..250]|IsEven(k) and Factorial(k) mod Binomial(k+1,2) eq 0]; // Marius A. Burtea, Jul 28 2019

Mathematica

Complement[Table[2 n, {n, 1, 103}], Table[EulerPhi[Prime[n]], {n, 1, 103}]] (* Terry D. Grant, Oct 31 2020 *)

PARI

forcomposite(c=4,10^3,if(c%2==1,print1(c-1,", "))); \\ Joerg Arndt, Jul 25 2019

Python

from sympy import primepi
def A309355(n):
    if n == 1: return 8
    m, k = n, primepi(n) + n + (n>>1)
    while m != k:
        m, k = k, primepi(k) + n + (k>>1)
    return m-1 # Chai Wah Wu, Aug 02 2024

Formula

a(n) = A071904(n) - 1.