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A383863 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.

%I A383863 #10 May 16 2025 18:19:44
%S A383863 1,2,2,3,2,4,2,3,3,4,2,6,2,4,4,3,2,6,2,6,4,4,2,6,3,4,3,6,2,8,2,3,4,4,
%T A383863 4,9,2,4,4,6,2,8,2,6,6,4,2,6,3,6,4,6,2,6,4,6,4,4,2,12,2,4,6,5,4,8,2,6,
%U A383863 4,8,2,9,2,4,6,6,4,8,2,6,3,4,2,12,4,4,4
%N A383863 The number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a unitary divisor of the p-adic valuation of n.
%C A383863 First differs from A073184 at n = 64.
%C A383863 First differs from A383865 at n = 256.
%C A383863 The number of divisors d of n such that each is a unitary divisor of an exponential unitary divisor of n (see A361255).	
%C A383863 Analogous to the number of (1+e)-divisors (A049599) as exponential unitary divisors (A361255, A278908) are analogous to exponential divisors (A322791, A049419).
%C A383863 The sum of these divisors is A383864(n).
%C A383863 Also, the number of divisors d of n having the property that for every prime p dividing n the p-adic valuation of d is either 0 or a squarefree divisor of the p-adic valuation of n. The sum of these divisors is A383867(n).
%H A383863 Amiram Eldar, <a href="/A383863/b383863.txt">Table of n, a(n) for n = 1..10000</a>
%F A383863 Multiplicative with a(p^e) = 1 + 2^A001221(e) = 1 + A034444(e).
%F A383863 a(n) <= A049599(n), with equality if and only if n is an exponentially squarefree number (A209061).
%t A383863 f[p_, e_] := 2^PrimeNu[e] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A383863 (PARI) a(n) = vecprod(apply(x -> 1 + 1 << omega(x), factor(n)[, 2]));
%Y A383863 Cf. A001221, A034444, A049419, A049599, A073184, A209061, A278908, A322791, A361255, A383864, A383865, A383867.
%K A383863 nonn,easy,mult
%O A383863 1,2
%A A383863 _Amiram Eldar_, May 12 2025