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%I #17 Dec 18 2023 01:41:52
%S 1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,6,1,13,1,1,1,1,1,1,1,
%T 1,4,1,1,1,1,1,1,1,1,4,1,1,1,8,6,1,1,1,13,1,1,1,1,1,1,1,1,4,1,1,1,1,1,
%U 1,1,1,4,1,1,6,1,1,1,1,1,40,1,1,1,1,1,1,1,1,4,1,1,1,1,1,1,1,8,4,6,1,1,1,1,1
%N Sum of divisors of A336651(n) (odd part of n divided by its largest squarefree divisor).
%H Antti Karttunen, <a href="/A336649/b336649.txt">Table of n, a(n) for n = 1..65537</a>
%F Multiplicative with a(2^e) = 1, a(p^1) = 1 and a(p^e) = (p^e - 1)/(p-1) if e > 1.
%F a(n) = A000203(A336651(n)) = A335341(A000265(n)).
%F a(n) = A336652(n) / A204455(n).
%F Dirichlet g.f.: zeta(s-1) * zeta(s) * (1 - 1/(1-2^s+2^(2*s-1))) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - _Amiram Eldar_, Dec 18 2023
%t f[2, e_] := 1; f[p_, e_] := (p^e - 1)/(p-1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Dec 07 2020 *)
%o (PARI) A336649(n) = { my(f=factor(n)); prod(i=1, #f~, if((2==f[i,1])||(1==f[i,2]),1,(((f[i,1]^(f[i,2]))-1) / (f[i,1]-1)))); };
%o (PARI)
%o A000265(n) = (n>>valuation(n,2));
%o A335341(n) = if(1==n,n,sigma(n/factorback(factorint(n)[, 1])));
%o A336649(n) = A335341(A000265(n));
%Y Cf. A000203, A000265, A204455, A335341, A336651, A336652.
%K nonn,mult
%O 1,9
%A _Antti Karttunen_, Jul 30 2020
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