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A383867 The sum 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.

%I A383867 #11 May 17 2025 08:15:25
%S A383867 1,3,4,7,6,12,8,11,13,18,12,28,14,24,24,7,18,39,20,42,32,36,24,44,31,
%T A383867 42,31,56,30,72,32,35,48,54,48,91,38,60,56,66,42,96,44,84,78,72,48,28,
%U A383867 57,93,72,98,54,93,72,88,80,90,60,168,62,96,104,79,84,144,68
%N A383867 The sum 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.
%C A383867 Analogous to the sum of (1+e)-divisors (A051378) as exponential squarefree exponential divisors (A383761, A361174) are analogous to exponential divisors (A322791, A051377).
%C A383867 The number of these divisors is A383863(n).
%H A383867 Amiram Eldar, <a href="/A383867/b383867.txt">Table of n, a(n) for n = 1..10000</a>
%F A383867 Multiplicative with a(p^e) = 1 + Sum_{d squarefree divisor of e} p^d.
%F A383867 a(n) <= A051378(n), with equality if and only if n is an exponentially squarefree number (A209061).
%F A383867 Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} f(1/p) = 1.47709589136345836345..., and f(x) = (1-x) * (1 + Sum_{k>=1} (1 + Sum{d|k, d squarefree} x^(2*k-d))).
%t A383867 f[p_, e_] := 1 + DivisorSum[e, p^# &, SquareFreeQ[#] &]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A383867 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, 1 + sumdiv(f[i, 2], d, if(issquarefree(d), f[i, 1]^d)));}
%Y A383867 Cf. A051378, A209061, A322791, A361174, A383761, A383863.
%K A383867 nonn,easy,mult
%O A383867 1,2
%A A383867 _Amiram Eldar_, May 13 2025