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A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).

%I A384558 #10 Jun 04 2025 06:57:48
%S A384558 1,2,3,2,5,6,7,10,3,10,11,6,13,14,15,2,17,6,19,10,21,22,23,30,5,26,30,
%T A384558 14,29,30,31,34,33,34,35,6,37,38,39,50,41,42,43,22,15,46,47,6,7,10,51,
%U A384558 26,53,60,55,70,57,58,59,30,61,62,21,10,65,66,67,34,69
%N A384558 The sum of the exponential divisors of n that are exponentially odd numbers (A268335).
%C A384558 First differs from A384559 at n = 512: a(512) = 522, while A384559(512) = 514.
%C A384558 The number of these divisors is A368979(n), and the largest of them is A331737(n).
%C A384558 The indices of records of a(n)/n are the primorial numbers (A002110) cubed, i.e., 1 and the terms of A115964.
%H A384558 Amiram Eldar, <a href="/A384558/b384558.txt">Table of n, a(n) for n = 1..10000</a>
%F A384558 Multiplicative with a(p^e) = Sum_{d|e, d odd} p^d.
%F A384558 a(n) = n if and only if n is squarefree (A005117).
%F A384558 a(n) < n if and only if n is in A072587.
%F A384558 a(n) > n if and only if n is in A374459.
%F A384558 limsup_{n->oo} a(n)/n = Product_{p prime} (1 + 1/p^2) = 15/Pi^2 (A082020).
%F A384558 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p*(p^2-1)) - 1/(p^2-1) + (1-1/p) * Sum_{k>=1} p^(2*k+1)/(p^(4*k+2)-1)) = 0.80824764393216997768... .
%p A384558 A384558:=proc(n)
%p A384558     local a, pe,p,e,af,d;
%p A384558     a := 1;
%p A384558     for pe in ifactors(n)[2] do
%p A384558         p := op(1,pe) ;
%p A384558         e := op(2,pe) ;
%p A384558         af := 0 ;
%p A384558         for d in numtheory[divisors](e) do
%p A384558             if type(d,'odd') then
%p A384558                 af := af+p^d ;
%p A384558             end if;
%p A384558         end do:
%p A384558         a := a*af ;
%p A384558     end do;
%p A384558     a
%p A384558 end proc:
%p A384558 seq(A384558(n), n=1..100); # _R. J. Mathar_, Jun 04 2025
%t A384558 f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384558 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, (d % 2) * f[i,1]^d));}
%Y A384558 Cf. A002110, A005117, A051377, A072587,  A082020, A115964, A268335, A331737, A368979, A374459, A384559.
%K A384558 nonn,easy,mult
%O A384558 1,2
%A A384558 _Amiram Eldar_, Jun 03 2025