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A368978 The number of bi-unitary divisors of n that are squares (A000290).

%I A368978 #9 Jan 12 2024 22:48:38
%S A368978 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,3,1,1,
%T A368978 1,4,1,1,1,2,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,4,1,1,1,2,
%U A368978 1,1,1,4,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1
%N A368978 The number of bi-unitary divisors of n that are squares (A000290).
%C A368978 First differs from A007424, A278908, A307848, A323308, A358260 and A365549 at n = 32.
%H A368978 Amiram Eldar, <a href="/A368978/b368978.txt">Table of n, a(n) for n = 1..10000</a>
%F A368978 Multiplicative with a(p^e) = (e + 1)/2 if e is odd, and 2*floor((e+2)/4) if e is even.
%F A368978 a(n) >= 1, with equality if and only if n is squarefree (A005117).
%F A368978 a(n) <= A286324(n), with equality if and only if n is in A062503.
%F A368978 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(4) * Product_{p prime} (1 + 1/p^2 - 1/p^4 + 1/p^5) = 1.58922450321701775833... .
%t A368978 f[p_, e_] := If[OddQ[e], (e + 1)/2, 2*Floor[(e+2)/4]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A368978 (PARI) a(n) = vecprod(apply(x -> if(x%2, (x+1)/2, 2*((x+2)\4)), factor(n)[, 2]));
%Y A368978 Cf. A000005, A000290, A005117, A013662, A062503, A222266, A286324, A368977.
%Y A368978 Cf. A007424, A278908, A307848, A323308, A365549.
%Y A368978 Similar sequences: A046951, A056624, A358260, A368980.
%K A368978 nonn,easy,mult
%O A368978 1,4
%A A368978 _Amiram Eldar_, Jan 11 2024