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

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, 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, 1, 1, 1, 4, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1

Offset

1,4

Comments

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

The number of exponential divisors (A322791) of n that are exponentially squarefree numbers (A209061). - Amiram Eldar, Jan 13 2026

Links

Antti Karttunen, Table of n, a(n) for n = 1..10000

Xiaodong Cao and Wenguang Zhai, Some arithmetic functions involving exponential divisors, Journal of Integer Sequences, Vol. 13 (2010), Article 10.3.7, eq (20).

Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Y. F. S. Pétermann, Arithmetical functions involving exponential divisors: Note on two papers by L. Tóth, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 32 (2010), pp. 143-149. See Theorem A, p. 144.

László Tóth, On certain arithmetic functions involving exponential divisors, II, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 27 (2007), pp. 155-166; arXiv preprint, arXiv:0708.3557 [math.NT], 2007-2009.

Xiangzhen Zhao, Min Liu, and Yu Huang, Mean value for the function t^(e)(n) over square-full numbers, Scientia Magna, Vol. 8, No. 3 (2012), pp. 110-114.

Index entries for sequences computed from exponents in factorization of n.

Formula

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... (A393489) (Tóth, 2007). - Amiram Eldar, Nov 08 2020

More precisely, Sum_{k=1..n} a(k) = C_1 * n + C_2 * sqrt(n) + O(n^(1/4+eps)), where C_1 = A393489 and C_2 = zeta(1/2) * Product_{p prime} (1 + Sum_{k>=4} (2^omega(k) - 2^omega(k-1) - 2^omega(k-2) + 2^omega(k-3))/p^(k/2)) = -0.99308863474918955038... (Pétermann, 2010). - Amiram Eldar, Mar 17 2026

Maple

A278908 := proc(n)
    local a, p, e;
    a := 1;
    if n =1 then
        ;
    else
        for p in ifactors(n)[2] do
            e := op(2, p) ;
            a := a*2^A001221(e) ;
        end do:
    end if;
    a ;
end proc:

Mathematica

Table[Times @@ Apply[Times, FactorInteger[n] /. {p_, e_} /; p > 1 :> 2^PrimeNu[e]], {n, 105}] (* Michael De Vlieger, Jul 29 2017 *)

Prog

(Scheme) (define (A278908 n) (if (= 1 n) n (* (A000079 (A001221 (A067029 n))) (A278908 (A028234 n))))) ;; Antti Karttunen, Jul 27 2017
(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
(Python)
from sympy.functions.combinatorial.numbers import primenu, factorint
def A278908(n): return 1<<sum(map(primenu, factorint(n).values())) # Chai Wah Wu, Oct 18 2025

Crossrefs

Cf. A001221, A059750, A209061, A322791, A393489.

Sequence in context: A390490 A323308 A365549 * A307848 A358260 A368978

Adjacent sequences: A278905 A278906 A278907 * A278909 A278910 A278911

Keyword

nonn,easy,mult

Author

R. J. Mathar, Nov 30 2016

Status

approved