cp's OEIS Frontend

A351568 Sum of the divisors of the largest unitary divisor of n that is a square.

%I A351568 #20 Sep 03 2023 10:45:37
%S A351568 1,1,1,7,1,1,1,1,13,1,1,7,1,1,1,31,1,13,1,7,1,1,1,1,31,1,1,7,1,1,1,1,
%T A351568 1,1,1,91,1,1,1,1,1,1,1,7,13,1,1,31,57,31,1,7,1,1,1,1,1,1,1,7,1,1,13,
%U A351568 127,1,1,1,7,1,1,1,13,1,1,31,7,1,1,1,31,121,1,1,7,1,1,1,1,1,13,1,7,1,1,1,1,1,57,13
%N A351568 Sum of the divisors of the largest unitary divisor of n that is a square.
%C A351568 Obviously, all terms are odd.
%H A351568 Antti Karttunen, <a href="/A351568/b351568.txt">Table of n, a(n) for n = 1..65537</a>
%H A351568 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A351568 Multiplicative with a(p^e) = (p^(e+1)-1)/(p-1) if e is even and 1 otherwise.
%F A351568 a(n) = A000203(A350388(n)).
%F A351568 a(n) = A000203(n) / A351569(n).
%F A351568 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = (zeta(3)/3) * Product_{p prime} (1 + 1/p^(3/2) + 1/p^2 - 1/p^(5/2)) = 1.008259499413... . - _Amiram Eldar_, Nov 20 2022
%F A351568 Dirichlet g.f.: zeta(2*s) * zeta(2*s-2) * Product_{p prime} (1 + 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-2)). - _Amiram Eldar_, Sep 03 2023
%t A351568 f[p_, e_] := If[EvenQ[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 A351568 (PARI)
%o A351568 A350388(n) = { my(m=1, f=factor(n)); for(k=1,#f~,if(0==(f[k,2]%2), m *= (f[k,1]^f[k,2]))); (m); };
%o A351568 A351568(n) = sigma(A350388(n));
%o A351568 (Python)
%o A351568 from math import prod
%o A351568 from sympy import factorint
%o A351568 def A351568(n): return prod(1 if e % 2 else (p**(e+1)-1)//(p-1) for p, e in factorint(n).items()) # _Chai Wah Wu_, Feb 24 2022
%Y A351568 Cf. A000203, A002117, A350388, A351569, A351570, A351575 (positions of primes).
%K A351568 nonn,mult
%O A351568 1,4
%A A351568 _Antti Karttunen_, Feb 23 2022