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A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.

%I A356242 #11 Feb 16 2025 08:34:03
%S A356242 0,0,1,0,1,1,0,0,2,1,0,1,0,0,2,0,1,2,0,1,1,0,0,1,2,0,3,0,0,2,0,0,1,1,
%T A356242 1,2,0,0,1,1,0,1,0,0,3,0,0,1,0,2,2,0,0,3,1,0,1,0,0,2,0,0,2,0,1,1,0,1,
%U A356242 1,1,0,2,0,0,3,0,0,1,0,1,4,0,0,1,2,0,1
%N A356242 a(n) is the number of Fermat numbers dividing n, counted with multiplicity.
%C A356242 The multiplicity of a divisor d (not necessarily a prime) of n is defined in A169594 (see also the first formula).
%C A356242 A000244(n) is the least number k such that a(k) = n.
%C A356242 The asymptotic density of occurrences of 0 is 1/2.
%C A356242 The asymptotic density of occurrences of 1 is (1/2) * Sum_{k>=0} 1/(2^(2^k)+1) = (1/2) * A051158 = 0.2980315860... .
%H A356242 Amiram Eldar, <a href="/A356242/b356242.txt">Table of n, a(n) for n = 1..10000</a>
%H A356242 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FermatNumber.html">Fermat Number</a>.
%H A356242 Wikipedia, <a href="http://en.wikipedia.org/wiki/Fermat_number">Fermat number</a>.
%F A356242 a(n) = Sum_{k>=1} v(A000215(k), n), where v(m, n) is the exponent of the largest power of m that divides n.
%F A356242 a(A000215(n)) = 1.
%F A356242 a(A000244(n)) = a(A000351(n)) = a(A001026(n)) = n.
%F A356242 a(A003593(n)) = A112754(n).
%F A356242 a(n) >= A356241(n).
%F A356242 a(A051179(n)) = n.
%F A356242 a(A080307(n)) > 0 and a(A080308(n)) = 0.
%F A356242 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{k>=0} 1/(2^(2^k)) = 0.8164215090... (A007404).
%t A356242 f = Table[(2^(2^n) + 1), {n, 0, 5}]; a[n_] := Total[IntegerExponent[n, f]]; Array[a, 100]
%Y A356242 Cf. A000244, A000215, A007404, A051158, A051179, A169594, A356241.
%Y A356242 Cf. A080307 (positions of nonzeros), A080308 (positions of 0's).
%K A356242 nonn
%O A356242 1,9
%A A356242 _Amiram Eldar_, Jul 30 2022