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A351569 Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.

%I A351569 #20 Sep 03 2023 10:45:46
%S A351569 1,3,4,1,6,12,8,15,1,18,12,4,14,24,24,1,18,3,20,6,32,36,24,60,1,42,40,
%T A351569 8,30,72,32,63,48,54,48,1,38,60,56,90,42,96,44,12,6,72,48,4,1,3,72,14,
%U A351569 54,120,72,120,80,90,60,24,62,96,8,1,84,144,68,18,96,144,72,15,74,114,4,20,96,168,80,6,1,126,84
%N A351569 Sum of divisors of the largest unitary divisor of n that is an exponentially odd number.
%H A351569 Antti Karttunen, <a href="/A351569/b351569.txt">Table of n, a(n) for n = 1..20000</a>
%H A351569 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A351569 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is odd and 1 otherwise.
%F A351569 a(n) = A000203(A350389(n)).
%F A351569 a(n) = A000203(n) / A351568(n).
%F A351569 Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(4)/2 = Pi^4/180 = 0.541161... . - _Amiram Eldar_, Nov 20 2022
%F A351569 Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^s - 1/p^(2*s-2)). - _Amiram Eldar_, Sep 03 2023
%t A351569 f[p_, e_] := If[OddQ[e], (p^(e + 1) - 1)/(p - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Feb 23 2022 *)
%o A351569 (PARI)
%o A351569 A350389(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(1==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
%o A351569 A351569(n) = sigma(A350389(n));
%o A351569 (Python)
%o A351569 from math import prod
%o A351569 from sympy import factorint
%o A351569 def A351569(n): return prod((p**(e+1)-1)//(p-1) if e % 2 else 1 for p, e in factorint(n).items()) # _Chai Wah Wu_, Feb 24 2022
%Y A351569 Cf. A000203, A013662, A028982 (positions of odd terms), A268335 (exponentially odd numbers), A350389, A351568, A351571.
%Y A351569 Coincides with A001615 on squarefree numbers, A005117.
%K A351569 nonn,mult
%O A351569 1,2
%A A351569 _Antti Karttunen_, Feb 23 2022