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A278908 Multiplicative with a(p^e) = 2^omega(e), where omega = A001221.

%I A278908 #35 May 30 2025 03:50:20
%S A278908 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T A278908 1,4,1,1,1,2,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,4,1,1,1,2,
%U A278908 1,1,1,4,1,1,2,2,1,1,1,2,2,1,1,2,1,1
%N A278908 Multiplicative with a(p^e) = 2^omega(e), where omega = A001221.
%C A278908 The number of exponential unitary (or e-unitary) divisors of n and the number of exponential squarefree exponential divisors (or e-squarefree e-divisors) of n. These are divisors of n = Product p(i)^a(i) of the form Product p(i)^b(i) where each b(i) is a unitary divisor of a(i) in the first case, or each b(i) is a squarefree divisor of a(i) in the second case. - _Amiram Eldar_, Dec 29 2018
%H A278908 Antti Karttunen, <a href="/A278908/b278908.txt">Table of n, a(n) for n = 1..10000</a>
%H A278908 Xiaodong Cao and Wenguang Zahi, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Cao/cao4.html">Some arithmetic functions involving exponential divisors</a>, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7, eq (20).
%H A278908 Nicuşor Minculete and László Tóth, <a href="https://doi.org/10.71352/ac.35.205">Exponential unitary divisors</a>, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.
%H A278908 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 A278908 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 A278908 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%F A278908 Asymptotic mean: lim_{n->oo} (1/n) * Sum_{k=1..n} a(k) = Product_{p prime} (1 + Sum_{k>=2} (2*omega(k) - 2^omega(k-1))/p^k) = 1.5431653193... (Tóth, 2007). - _Amiram Eldar_, Nov 08 2020
%p A278908 A278908 := proc(n)
%p A278908     local a,p,e;
%p A278908     a := 1;
%p A278908     if n =1 then
%p A278908         ;
%p A278908     else
%p A278908         for p in ifactors(n)[2] do
%p A278908             e := op(2,p) ;
%p A278908             a := a*2^A001221(e) ;
%p A278908         end do:
%p A278908     end if;
%p A278908     a ;
%p A278908 end proc:
%t A278908 Table[Times @@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> 2^PrimeNu[e]], {n, 105}] (* _Michael De Vlieger_, Jul 29 2017 *)
%o A278908 (Scheme) (define (A278908 n) (if (= 1 n) n (* (A000079 (A001221 (A067029 n))) (A278908 (A028234 n))))) ;; _Antti Karttunen_, Jul 27 2017
%o A278908 (PARI) a(n) = my(f=factor(n)); for (k=1, #f~, f[k,1] = 2^omega(f[k,2]); f[k,2] = 1); factorback(f); \\ _Michel Marcus_, Jul 28 2017
%Y A278908 Cf. A001221.
%K A278908 nonn,easy,mult
%O A278908 1,4
%A A278908 _R. J. Mathar_, Nov 30 2016