A374671 - OEIS

%I #9 Jul 18 2024 14:27:28

%S 8,19,23,44,45,57,67,76,80,83,84,85,105,107,116,120,123,140,141,146,

%T 158,161,165,174,177,187,201,208,214,225,235,239,241,243,244,246,247,

%U 263,269,272,277,284,297,309,315,321,322,325,337,339,341,342,344,360,363

%N Positive numbers k such that k! and (k+1)! have an equal number of infinitary divisors.

%C Positive numbers such that k! and (k+1)! have an equal number of Fermi-Dirac factors (A064547).

%C Positive numbers k such that A037445(k!) = A037445((k+1)!).

%C Positive numbers k such that A064547(k!) = A064547((k+1)!).

%C Positive numbers k such that A177329(k) = A177329(k+1).

%H Amiram Eldar, <a href="/A374671/b374671.txt">Table of n, a(n) for n = 1..10000</a>

%e 8 is a term since A037445(8!) = A037445(9!) = 64.

%t s[n_] := s[n] = Module[{e = FactorInteger[n!][[;; , 2]]}, Sum[DigitCount[e[[k]], 2, 1], {k, 1, Length[e]}]]; Select[Range[2, 400], 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, A374672, A374673, A374674.

%K nonn

%O 1,1

%A _Amiram Eldar_, Jul 16 2024