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A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.

%I A306069 #18 Dec 29 2021 05:58:24
%S A306069 1,3,5,7,9,13,15,19,21,25,27,31,33,37,41,45,47,51,53,57,61,65,67,75,
%T A306069 77,81,85,89,91,99,101,107,111,115,119,123,125,129,133,141,143,151,
%U A306069 153,157,161,165,167,175,177,181,185,189,191,199,203,211,215,219,221
%N A306069 Partial sums of A286324: Sum_{k=1..n} bd(k) where bd(k) is the number of bi-unitary divisors of k.
%C A306069 The bi-unitary version of A006218 and A064608.
%D A306069 József Sándor, Dragoslav S. Mitrinovic, Borislav Crstici, Handbook of Number Theory I, Springer Science & Business Media, 2005, page 72.
%H A306069 Amiram Eldar, <a href="/A306069/b306069.txt">Table of n, a(n) for n = 1..10000</a>
%H A306069 D. Suryanarayana, <a href="https://doi.org/10.1007/BFb0058797">The number of bi-unitary divisors of an integer</a>, The theory of arithmetic functions, ed. Anthony A. Gioia and Donald L. Goldsmith, Springer, Berlin, Heidelberg, 1972, pp. 273-282.
%H A306069 D. Suryanarayana and R. Sita Rama Chandra Rao, <a href="http://informaticsjournals.com/index.php/jims/article/view/16651">The number of bi-unitary divisors of an integer - II</a>, Journal of the Indian mathematical Society, Vol. 39, No. 1-4 (1975), pp. 261-280.
%F A306069 a(n) = A*n*(log(n) + 2*gamma - 1 + B) + O(n^(1/2)*exp(-A * log(n)^(3/5) * log(log(n))^(-1/5))), where gamma = A001620, A = A306071 and B = A306072.
%t A306069 fun[p_, e_] := If[Mod[e, 2] == 1, (e + 1), e]; bdivnum[n_] := If[n==1,1,Times @@ (fun @@@ FactorInteger[n])]; Accumulate@ Array[bdivnum, {60}]
%o A306069 (PARI) udivs(n) = {my(d = divisors(n)); select(x->(gcd(x, n/x)==1), d); }
%o A306069 gcud(n, m) = vecmax(setintersect(udivs(n), udivs(m)));
%o A306069 biudivs(n) = select(x->(gcud(x, n/x)==1), divisors(n));
%o A306069 a(n) = sum(k=1, n, #biudivs(k)); \\ _Michel Marcus_, Jun 20 2018
%Y A306069 Cf. A001620, A006218, A064608, A286324, A306071, A306072.
%K A306069 nonn
%O A306069 1,2
%A A306069 _Amiram Eldar_, Jun 19 2018