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A383763 The sum of unitary divisors of n that are exponentially squarefree numbers.

%I A383763 #9 May 11 2025 01:19:23
%S A383763 1,3,4,5,6,12,8,9,10,18,12,20,14,24,24,1,18,30,20,30,32,36,24,36,26,
%T A383763 42,28,40,30,72,32,33,48,54,48,50,38,60,56,54,42,96,44,60,60,72,48,4,
%U A383763 50,78,72,70,54,84,72,72,80,90,60,120,62,96,80,65,84,144,68
%N A383763 The sum of unitary divisors of n that are exponentially squarefree numbers.
%C A383763 The number of these divisors is A383762(n) and the largest of them is A383764(n).
%H A383763 Amiram Eldar, <a href="/A383763/b383763.txt">Table of n, a(n) for n = 1..10000</a>
%F A383763 Multiplicative with a(p^e) = p^e + 1 if e is squarefree (A005117), and 1 otherwise.
%F A383763 a(n) <= A034448(n), with equality if and only if n is an exponentially squarefree number (A209061).
%F A383763 a(n) <= A365682(n), with equality if and only if n is a squarefree number.
%t A383763 f[p_, e_] := If[SquareFreeQ[e], p^e + 1, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A383763 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(issquarefree(f[i,2]), f[i,1]^f[i,2]+1, 1));}
%Y A383763 Cf. A005117, A034448, A077610, A209061, A365682, A383762, A383764.
%K A383763 nonn,easy,mult
%O A383763 1,2
%A A383763 _Amiram Eldar_, May 09 2025