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

%I A309355 #42 Aug 02 2024 11:30:32
%S A309355 8,14,20,24,26,32,34,38,44,48,50,54,56,62,64,68,74,76,80,84,86,90,92,
%T A309355 94,98,104,110,114,116,118,120,122,124,128,132,134,140,142,144,146,
%U A309355 152,154,158,160,164,168,170,174,176,182,184,186,188,194,200,202,204,206
%N A309355 Even numbers k such that k! is divisible by k*(k+1)/2.
%C A309355 Even terms in A060462.
%C A309355 And A071904 are the successors of a(n).
%C A309355 Even numbers that are not a prime - 1. That is, even numbers not in A006093. - _Terry D. Grant_, Oct 31 2020
%D A309355 J. D. E. Konhauser et al., Which Way Did The Bicycle Go?, Problem 98, pp. 29; 145-146, MAA Washington DC, 1996.
%D A309355 Die WURZEL - Zeitschrift für Mathematik, 53. Jahrgang, Juli 2019, S. 171, WURZEL-Aufgabe 2019-36 von Gerhard Dietel, Regensburg.
%F A309355 a(n) = A071904(n) - 1.
%e A309355 8! = 40320 is divisible by 8*9/2 = 36.
%e A309355 14! is divisible by 14*15/2.
%t A309355 Complement[Table[2 n, {n, 1, 103}], Table[EulerPhi[Prime[n]], {n, 1, 103}]] (* _Terry D. Grant_, Oct 31 2020 *)
%o A309355 (PARI) forcomposite(c=4,10^3,if(c%2==1,print1(c-1,", "))); \\ _Joerg Arndt_, Jul 25 2019
%o A309355 (Magma) [k: k in [2..250]|IsEven(k) and Factorial(k) mod Binomial(k+1,2) eq 0]; // _Marius A. Burtea_, Jul 28 2019
%o A309355 (Python)
%o A309355 from sympy import primepi
%o A309355 def A309355(n):
%o A309355     if n == 1: return 8
%o A309355     m, k = n, primepi(n) + n + (n>>1)
%o A309355     while m != k:
%o A309355         m, k = k, primepi(k) + n + (k>>1)
%o A309355     return m-1 # _Chai Wah Wu_, Aug 02 2024
%Y A309355 Cf. A060462, A071904.
%Y A309355 Essentially the same as A186193.
%Y A309355 Cf. A006093.
%K A309355 nonn
%O A309355 1,1
%A A309355 _Gerhard Palme_, Jul 25 2019