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A309355
Even numbers k such that k! is divisible by k*(k+1)/2.
1
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
1,1
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
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.
FORMULA
a(n) = A071904(n) - 1.
EXAMPLE
8! = 40320 is divisible by 8*9/2 = 36.
14! is divisible by 14*15/2.
MATHEMATICA
Complement[Table[2 n, {n, 1, 103}], Table[EulerPhi[Prime[n]], {n, 1, 103}]] (* Terry D. Grant, Oct 31 2020 *)
PROG
(PARI) forcomposite(c=4, 10^3, if(c%2==1, print1(c-1, ", "))); \\ Joerg Arndt, Jul 25 2019
(Magma) [k: k in [2..250]|IsEven(k) and Factorial(k) mod Binomial(k+1, 2) eq 0]; // Marius A. Burtea, Jul 28 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
CROSSREFS
Essentially the same as A186193.
Cf. A006093.
Sequence in context: A025044 A264722 A125163 * A374223 A063288 A136798
KEYWORD
nonn
AUTHOR
Gerhard Palme, Jul 25 2019
STATUS
approved