cp's OEIS Frontend

A361174 The sum of the exponential squarefree exponential divisors (or e-squarefree e-divisors) of n.

%I A361174 #21 May 30 2025 08:02:00
%S A361174 1,2,3,6,5,6,7,10,12,10,11,18,13,14,15,6,17,24,19,30,21,22,23,30,30,
%T A361174 26,30,42,29,30,31,34,33,34,35,72,37,38,39,50,41,42,43,66,60,46,47,18,
%U A361174 56,60,51,78,53,60,55,70,57,58,59,90,61,62,84,78,65,66,67,102
%N A361174 The sum of the exponential squarefree exponential divisors (or e-squarefree e-divisors) of n.
%C A361174 The exponential squarefree exponential divisors (or e-squarefree e-divisors) of n = Product_i p(i)^e(i) are all the numbers of the form Product_i p(i)^d(i) where d(i) is a squarefree divisor of e(i).
%C A361174 The number of exponential squarefree exponential divisors of n is A278908(n).
%H A361174 Amiram Eldar, <a href="/A361174/b361174.txt">Table of n, a(n) for n = 1..10000</a>
%H A361174 László Tóth, <a href="https://ac.inf.elte.hu/Vol_027_2007/doi/155_27.pdf">On certain arithmetic functions involving exponential divisors, II</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; <a href="https://arxiv.org/abs/0708.3557">arXiv preprint</a>, arXiv:0708.3557 [math.NT], 2007-2009.
%H A361174 Xiangzhen Zhao, Min Liu, and Yu Huang, <a href="http://fs.unm.edu/ScientiaMagna8no3.pdf#page=116">Mean value for the function t^(e)(n) over square-full numbers</a>, Scientia Magna, Vol. 8, No. 3 (2012), pp. 110-114.
%H A361174 <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>.
%F A361174 Multiplicative with a(p^e) = Sum_{d|e, d squarefree} p^d.
%F A361174 Sum_{k=1..n} a(k) ~ c * n^2 / 2, c = Product_{p prime} (1 + Sum_{k>=2} (a(p^k) - p*a(p^(k-1)))/p^(2*k)) = 1.08989220899432387559... . - _Amiram Eldar_, Feb 13 2024
%t A361174 f[p_, e_] := DivisorSum[e, p^# &, SquareFreeQ[#] &]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A361174 (PARI) ff(p, e) = sumdiv(e, d, if(issquarefree(d), p^d, 0));
%o A361174 a(n) = {my(f=factor(n)); prod(i=1, #f~, ff(f[i, 1], f[i, 2]));}
%Y A361174 Cf. A278908.
%Y A361174 Similar sequences: A051377, A322857, A323309, A361175.
%K A361174 nonn,mult
%O A361174 1,2
%A A361174 _Amiram Eldar_, Mar 03 2023