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Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).
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%I #25 Jan 04 2023 02:03:32

%S 1,1,4,1,6,4,8,1,12,6,12,4,14,8,24,1,18,12,20,6,32,12,24,4,30,14,36,8,

%T 30,24,32,1,48,18,48,12,38,20,56,6,42,32,44,12,72,24,48,4,56,30,72,14,

%U 54,36,72,8,80,30,60,24,62,32,96,1,84,48,68,18,96,48,72,12,74,38,120,20,96,56,80,6,108,42,84,32,108

%N Dedekind psi function applied to the odd part of n: a(n) = A001615(A000265(n)).

%C Coincides with A000593 on A122132.

%H Antti Karttunen, <a href="/A347385/b347385.txt">Table of n, a(n) for n = 1..20000</a>

%F Multiplicative with a(2^e) = 1, a(p^e) = (p+1)*p^(e-1) for all odd primes p.

%F a(n) = A001615(A000265(n)).

%F a(n) = A206787(n) * A336651(n). - _Antti Karttunen_, Feb 11 2022

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = 4/Pi^2 = 0.405284... (A185199). - _Amiram Eldar_, Nov 19 2022

%F Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s))*(4^s-2^(s+1))/(4^s-1). - _Amiram Eldar_, Jan 04 2023

%t f[p_, e_] := If[p == 2, 1, (p + 1)*p^(e - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Aug 31 2021 *)

%o (PARI) A347385(n) = if(1==n,n, my(f=factor(n>>valuation(n, 2))); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));

%Y Cf. A000265, A001615, A122132, A185199, A206787, A336651, A347386, A347387, A351035.

%Y Cf. also A000593, A053575.

%K nonn,mult

%O 1,3

%A _Antti Karttunen_, Aug 31 2021