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A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).

%I A133700 #27 Oct 28 2023 04:04:57
%S A133700 1,2,3,3,3,6,3,4,6,6,3,9,3,6,9,5,3,12,3,9,9,6,3,12,6,6,10,9,3,18,3,6,
%T A133700 9,6,9,18,3,6,9,12,3,18,3,9,18,6,3,15,6,12,9,9,3,20,9,12,9,6,3,27,3,6,
%U A133700 18,7,9,18,3,9,9,18,3,24,3,6,18,9,9,18,3,15,15,6,3,27,9,6,9,12,3,36,9,9,9
%N A133700 A051731 * A001227; a(n) = Sum_{d|n} A001227(d).
%H A133700 Antti Karttunen, <a href="/A133700/b133700.txt">Table of n, a(n) for n = 1..65537</a>
%F A133700 Inverse Möbius transform of A001227, the number of odd divisors of n. Row sums of triangle A133699.
%F A133700 Dirichlet g.f. (zeta(s))^3*(1-1/2^s). - _R. J. Mathar_, Feb 07 2011
%F A133700 a(n) = Sum_{d|n} A001227(d). - _Antti Karttunen_, Sep 27 2018
%F A133700 Sum_{k=1..n} a(k) ~ n/4 * (log(n)^2 + (6*g - 2 + 2*log(2))*log(n) + 2 + 6*g^2 - log(2)^2 - 2*log(2) + 6*g*(log(2) - 1) - 6*sg1), where g is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant A082633. - _Vaclav Kotesovec_, Feb 02 2019
%F A133700 G.f.: Sum_{k>=1} tau(k)*x^k/(1 - x^(2*k)), where tau = A000005. - _Ilya Gutkovskiy_, Sep 13 2019
%F A133700 Multiplicative with a(2^e) = e+1, and a(p^e) = (e+1)*(e+2)/2 for an odd prime p. - _Amiram Eldar_, Oct 28 2023
%e A133700 a(4) = sum of row 4 terms of triangle A133699: (1 + 1 + 0 + 1) = (1, 1, 0, 1) dot (1, 1, 2, 1), where A001227 = (1, 1, 2, 1, 2, 2, 2, 1, 3, ...).
%t A133700 f[p_, e_] := (e+1)*(e+2)/2; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 28 2023 *)
%o A133700 (PARI) A133700(n) = sumdiv(n,d,numdiv(d>>valuation(d,2))); \\ _Antti Karttunen_, Sep 27 2018
%Y A133700 Cf. A000005, A001227, A051731, A133699.
%Y A133700 Cf. A001620, A082633.
%K A133700 nonn,easy,mult
%O A133700 1,2
%A A133700 _Gary W. Adamson_, Sep 21 2007
%E A133700 More terms from _R. J. Mathar_, Jan 19 2009
%E A133700 Second, equivalent formula added to the definition by _Antti Karttunen_, Sep 27 2018