%I #7 Jul 18 2024 14:27:23

%S 5,9,17,27,33,34,35,43,48,51,53,59,65,68,69,75,77,87,91,97,98,99,103,

%T 115,119,125,129,134,135,139,147,149,151,155,163,164,171,179,183,189,

%U 194,195,197,199,203,211,215,221,227,229,230,231,237,245,249,257,259

%N Numbers k such that k! has more infinitary divisors than (k+1)!.

%C Numbers k such that k! has more Fermi-Dirac factors (A064547) than (k+1)!.

%C Numbers k such that A037445(k!) > A037445((k+1)!).

%C Numbers k such that A064547(k!) > A064547((k+1)!).

%C Numbers k such that A177329(k) > A177329(k+1).

%H Amiram Eldar, <a href="/A374672/b374672.txt">Table of n, a(n) for n = 1..10000</a>

%e 5 is a term since A037445(5!) = 16 > A037445(6!) = 8.

%t s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 300], s[#] > s[# + 1] &]

%o (PARI) s(n) = {my(e = factor(n!)[, 2]); sum(k=1, #e, hammingweight(e[k]));}

%o lista(kmax) = {my(s1 = s(1), s2); for(k = 2, kmax, s2 = s(k); if(s1 > s2, print1(k-1, ", ")); s1 = s2);}

%Y Cf. A037445, A064547, A177329, A343819, A374671, A374673, A374674.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jul 16 2024