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A356241 a(n) is the number of distinct Fermat numbers dividing n.

%I A356241 #12 Feb 16 2025 08:34:03
%S A356241 0,0,1,0,1,1,0,0,1,1,0,1,0,0,2,0,1,1,0,1,1,0,0,1,1,0,1,0,0,2,0,0,1,1,
%T A356241 1,1,0,0,1,1,0,1,0,0,2,0,0,1,0,1,2,0,0,1,1,0,1,0,0,2,0,0,1,0,1,1,0,1,
%U A356241 1,1,0,1,0,0,2,0,0,1,0,1,1,0,0,1,2,0,1
%N A356241 a(n) is the number of distinct Fermat numbers dividing n.
%C A356241 A051179(n) is the least number k such that a(k) = n.
%C A356241 The asymptotic density of occurrences of 0 is 1/2.
%C A356241 The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/2^(2^k) = (1/2) * A007404 = 0.4082107545... .
%H A356241 Amiram Eldar, <a href="/A356241/b356241.txt">Table of n, a(n) for n = 1..10000</a>
%H A356241 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.
%H A356241 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>.
%F A356241 a(A000215(n)) = 1.
%F A356241 a(A051179(n)) = n.
%F A356241 a(A003593(n)) = A112753(n).
%F A356241 a(n) <= A356242(n).
%F A356241 a(A080307(n)) > 0 and a(A080308(n)) = 0.
%F A356241 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)+1) = 0.5960631721... (A051158).
%t A356241 f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Count[f, _?(Divisible[n, #] &)]; Array[a, 100]
%Y A356241 Cf. A000215, A007404, A051158, A051179, A356242.
%Y A356241 Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).
%K A356241 nonn
%O A356241 1,15
%A A356241 _Amiram Eldar_, Jul 30 2022