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A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.

%I A372604 #9 May 08 2024 08:52:54
%S A372604 0,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,3,1,1,1,1,3,1,1,
%T A372604 1,1,1,1,1,3,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1,1,1,1,1,1,1,3,1,1,1,1,
%U A372604 1,1,1,3,1,1,1,1,1,1,1,3,3,1,1,1,1,1,1
%N A372604 The maximal exponent in the prime factorization of the largest divisor of n whose number of divisors is a power of 2.
%C A372604 First differs from A331273 at n = 32.
%C A372604 Differs from A368247 at n = 1, 128, 216, 256, 384, 432, 512, ... .
%C A372604 All the terms are of the form 2^k-1 (A000225).
%H A372604 Amiram Eldar, <a href="/A372604/b372604.txt">Table of n, a(n) for n = 1..10000</a>
%F A372604 a(n) = A051903(A372379(n)).
%F A372604 a(n) = A092323(A051903(n)+1).
%F A372604 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1 + Sum_{i>=1} 2^i * (1 - 1/zeta(2^(i+1)-1)) = 1.36955053734097783559... .
%e A372604 4 has 3 divisors, 1, 2 and 4. The number of divisors of 4 is 3, which is not a power of 2. The number of divisors of 2 is 2, which is a power of 2. Therefore, A372379(4) = 2 and a(4) = A051903(2) = 1.
%t A372604 f[n_] := 2^Floor[Log2[n + 1]] - 1; a[n_] := f[Max[FactorInteger[n][[;; , 2]]]]; a[1] = 0; Array[a, 100]
%o A372604 (PARI) s(n) = 2^exponent(n+1) - 1;
%o A372604 a(n) = if(n>1, s(vecmax(factor(n)[,2])), 0);
%Y A372604 Cf. A000225, A051903, A092323, A372379, A372466.
%Y A372604 Cf. A331273, A368247.
%Y A372604 Similar sequences: A007424, A368781, A372601, A372602, A372603.
%K A372604 nonn,easy
%O A372604 1,8
%A A372604 _Amiram Eldar_, May 07 2024