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%I A372331 #6 Apr 28 2024 11:35:24
%S A372331 1,1,1,2,1,1,1,1,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,1,2,1,1,2,1,1,1,2,1,1,
%T A372331 1,4,1,1,1,1,1,1,1,2,2,1,1,4,2,2,1,2,1,1,1,1,1,1,1,2,1,1,2,2,1,1,1,2,
%U A372331 1,1,1,2,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A372331 The number of infinitary divisors of the smallest number k such that k*n is a number whose number of divisors is a power of 2 (A036537).
%C A372331 First differs from A370077 and A370080 at n = 32.
%C A372331 The number of divisors d of n that are infinitarily relatively prime to n (see A064379), i.e., d have no common infinitary divisors with n.
%C A372331 Equivalently, the number of divisors d of n such that for each prime divisor p of d, bitand(v_p(n), v_p(d)) = 0, where v_p(k) is the highest power of p that divides k. Note that for infinitary divisors d of n (A077609), bitand(v_p(n), v_p(d)) = v_p(d).
%H A372331 Amiram Eldar, <a href="/A372331/b372331.txt">Table of n, a(n) for n = 1..10000</a>
%F A372331 a(n) = A037445(A372328(n)).
%F A372331 Multiplicative with a(p^e) = 2^A023416(e) = A080100(e).
%F A372331 a(n) = 1 if and only if n is in A036537.
%F A372331 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A080100(k)/p^k) = 1.51209151045338102469... .
%t A372331 f[p_, e_] := 2^DigitCount[e, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A372331 (PARI) a(n) = vecprod(apply(x -> 2^(logint(x, 2) + 1 - hammingweight(x)), factor(n)[, 2]));
%Y A372331 Cf. A036537, A037445, A064379, A077609, A080100, A372328.
%Y A372331 Cf. A370077, A370080.
%K A372331 nonn,easy,mult
%O A372331 1,4
%A A372331 _Amiram Eldar_, Apr 28 2024

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