cp's OEIS Frontend

A384559 The sum of the exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).

%I A384559 #9 Jun 03 2025 11:57:08
%S A384559 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 A384559 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 A384559 26,53,60,55,70,57,58,59,30,61,62,21,10,65,66,67,34,69
%N A384559 The sum of the exponential unitary (or e-unitary) divisors of n that are exponentially odd numbers (A268335).
%C A384559 First differs from A384558 at n = 512: a(512) = 514, while A384558(512) = 522.
%C A384559 The number of these divisors is A384557(n), and the largest of them is A331737(n).
%H A384559 Amiram Eldar, <a href="/A384559/b384559.txt">Table of n, a(n) for n = 1..10000</a>
%F A384559 Multiplicative with a(p^e) = Sum_{d|e, d odd, gcd(d, e/d) = 1} p^d.
%F A384559 a(n) = n if and only if n is squarefree (A005117).
%F A384559 a(n) < n if and only if n is in A072587.
%F A384559 a(n) > n if and only if n is in A374459.
%t A384559 f[p_, e_] := DivisorSum[e, p^# &, OddQ[#] && CoprimeQ[#, e/#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A384559 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, sumdiv(f[i,2], d, (d % 2) * (gcd(d, f[i,2]/d) == 1) * f[i,1]^d));}
%Y A384559 Cf. A005117, A051377, A072587, A268335, A331737, A368979, A374459, A384557, A384558.
%K A384559 nonn,easy,mult
%O A384559 1,2
%A A384559 _Amiram Eldar_, Jun 03 2025