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A306014 Numbers k such that A055228(k)^2 - A055228(k) is a multiple of k, where A055228(k) is ceiling(sqrt(k!)).

%I A306014 #62 Dec 04 2024 18:49:20
%S A306014 1,2,3,4,5,6,7,8,9,10,14,16,28,29,30,42,46,50,52,99,134,148,165,205,
%T A306014 245,249,315,390,441,461,525,560,763,962,1596,1666,1716,1847,1854,
%U A306014 1860,3515,4501,5179,6850,7345,7867,8940,9491,9523,15688,19988,23574,23956
%N A306014 Numbers k such that A055228(k)^2 - A055228(k) is a multiple of k, where A055228(k) is ceiling(sqrt(k!)).
%D A306014 Hazewinkel, Michiel, ed. (2001) [1994], Gamma Function, Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
%H A306014 Chai Wah Wu, <a href="/A306014/b306014.txt">Table of n, a(n) for n = 1..89</a> (all terms up to 10^6, n = 1..70 from Jon E. Schoenfield)
%e A306014 For k=6, A055228(6) = ceiling(sqrt(6!)) = 27, and 27^2-27 = 702, which is a multiple of 6.
%t A306014 Select[Range[4600],Divisible[Ceiling[Sqrt[#!]]^2-Ceiling[Sqrt[#!]],#]&] (* _Harvey P. Dale_, Mar 02 2023 *)
%o A306014 (PARI) default(realprecision,10^5); for(n=1,10^4, if( Mod( ceil(sqrt(n!)) - ceil(sqrt(n!))^2 , n) == 0, print1(n,", "))); \\ _Joerg Arndt_, Jun 17 2018
%Y A306014 Cf. A000188, A055228.
%K A306014 nonn
%O A306014 1,2
%A A306014 _Ivan Stoykov_, Jun 17 2018
%E A306014 Terms > 99 from _Joerg Arndt_, Jun 17 2018