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A365174 The sum of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).

%I A365174 #11 Mar 28 2025 02:21:57
%S A365174 1,3,4,7,6,12,8,15,13,18,12,28,14,24,24,27,18,39,20,42,32,36,24,60,31,
%T A365174 42,40,56,30,72,32,51,48,54,48,91,38,60,56,90,42,96,44,84,78,72,48,
%U A365174 108,57,93,72,98,54,120,72,120,80,90,60,168,62,96,104,107,84,144
%N A365174 The sum of divisors d of n such that gcd(d, n/d) is an exponentially odd number (A268335).
%C A365174 The number of these divisors is A365173(n).
%H A365174 Amiram Eldar, <a href="/A365174/b365174.txt">Table of n, a(n) for n = 1..10000</a>
%F A365174 Multiplicative with 1 + p^e + (p^(e + 1) - p)/(p^2 - 1) if e is even, and 1 + p^e + (1 + p^(2*floor(e/4)+1))*(p^(2*floor((e+1)/4)+1) - p)/(p^2 - 1) if e is odd.
%F A365174 a(n) <= A000203(n), with equality if and only if n is not a biquadrateful number (A046101).
%F A365174 a(n) >= A034448(n), with equality if and only if n is squarefree (A005117).
%F A365174 Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(6)/(2*zeta(3))) * Product_{p prime} (1 + 1/p^3 - 1/p^6) = 0.809912096042... .
%t A365174 f[p_, e_] := 1 + p^e + If[EvenQ[e], (p^(e + 1) - p)/(p^2 - 1), (1 + p^(2*Floor[e/4] + 1))*(p^(2*Floor[(e + 1)/4] + 1) - p)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A365174 (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; 1 + p^e + if(e%2, (1 + p^(2*(e\4) + 1))*(p^(2*((e+1)\4) + 1) - p)/(p^2 - 1), (p^(e+1)-p)/(p^2-1)));}
%Y A365174 Cf. A000203, A033634, A034448, A268335, A365172, A365173.
%Y A365174 Cf. A002117, A013661, A013664.
%K A365174 nonn,easy,mult
%O A365174 1,2
%A A365174 _Amiram Eldar_, Aug 25 2023