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A383865 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 an infinitary divisor of the p-adic valuation of n.

%I A383865 #7 May 16 2025 18:23:45
%S A383865 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 A383865 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 A383865 4,8,2,9,2,4,6,6,4,8,2,6,3,4,2,12,4,4,4
%N A383865 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 an infinitary divisor of the p-adic valuation of n.
%C A383865 First differs from A383863 at n = 256.
%C A383865 The number of divisors d of n such that each is a unitary divisor of an exponential infinitary divisor of n (see A383760).	
%C A383865 Analogous to the number of (1+e)-divisors (A049599) as exponential infinitary divisors (A383760, A307848) are analogous to exponential divisors (A322791, A049419).
%C A383865 The sum of these divisors is A383866(n).
%H A383865 Amiram Eldar, <a href="/A383865/b383865.txt">Table of n, a(n) for n = 1..10000</a>
%F A383865 Multiplicative with a(p^e) = 1 + A037445(e).
%F A383865 a(n) <= A049599(n), with equality if and only if all the exponents in the prime factorization of n are in A036537.
%t A383865 f[p_, e_] := 2^DigitCount[e, 2, 1]; d[1] = 1; d[n_] := Times @@ f @@@ FactorInteger[n]; ff[p_, e_] := d[e] + 1; a[1] = 1; a[n_] := Times @@ ff @@@ FactorInteger[n]; Array[a, 100]
%o A383865 (PARI) d(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n)[, 2]));
%o A383865 a(n) = vecprod(apply(x -> 1 + d(x), factor(n)[, 2]));
%Y A383865 Cf. A036537, A037445, A049419, A049599, A307848, A322791, A383760, A383863, A383866.
%K A383865 nonn,easy,mult
%O A383865 1,2
%A A383865 _Amiram Eldar_, May 12 2025