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A377562 Numbers that have twice as many infinitary divisors as noninfinitary divisors.

%I A377562 #10 Nov 01 2024 23:51:15
%S A377562 4,9,12,18,20,25,28,32,44,45,49,50,52,60,63,68,72,75,76,84,90,92,96,
%T A377562 98,99,108,116,117,121,124,126,132,140,147,148,150,153,156,160,164,
%U A377562 169,171,172,175,188,198,200,204,207,212,220,224,228,234,236,242,243,244,245
%N A377562 Numbers that have twice as many infinitary divisors as noninfinitary divisors.
%C A377562 Numbers k such that A037445(k) = 2 * A348341(k).
%C A377562 Numbers k such that A037445(k)/A000005(k) = 2/3. For numbers k in A036537, all the divisors are infinitary divisors, so A037445(k)/A000005(k) = 1. For numbers k that are not in A036537, the largest possible ratio A037445(k)/A000005(k) is 2/3.
%C A377562 Numbers whose prime factorization has exactly one exponent of the form 3*2^k-1, with k >= 0, and the rest of the exponents, if there are any, are of the form 2^k-1, with k >= 1.
%C A377562 The asymptotic density of this sequence is d * Sum_{p prime} (Sum_{k>=0} 1/p^(3*2^k-1))/(1 + Sum_{k>=1} 1/p^(2^k-1)) = 0.23171917985739378623..., where d = A327839.
%H A377562 Amiram Eldar, <a href="/A377562/b377562.txt">Table of n, a(n) for n = 1..10000</a>
%t A377562 f[p_, e_] := 2^DigitCount[e, 2, 1]/(e + 1); q[1] = False; q[n_] := Times @@ f @@@ FactorInteger[n] == 2/3; Select[Range[250], q]
%o A377562 (PARI) is(n) = {my(f = factor(n)); vecprod(apply(x -> (1 << hammingweight(x)) / (x+1), f[, 2])) == 2/3;}
%Y A377562 Cf. A000005, A000225, A036537, A037445, A060687 (unitary analog), A077609, A083329, A327839, A335832, A348341, A377563.
%K A377562 nonn,easy
%O A377562 1,1
%A A377562 _Amiram Eldar_, Nov 01 2024