A049599 -id:A049599 - cp's OEIS frontend

A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 8, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1

Offset: 1

Antti Karttunen, Sep 03 2018

The sequence seems to give the denominators of a few other similarly constructed rational valued sequences obtained as "Dirichlet Square Roots" (possibly of A282446 and A318469).

Antti Karttunen, Table of n, a(n) for n = 1..16385

Cf. A049599, A318671 (numerators), A318673.

Cf. also A046644, A299150, A317932, A317934, A318498.

up_to = (2^16)+1;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
v318671_62 = DirSqrt(vector(up_to, n, A049599(n)));
A318671(n) = numerator(v318671_62[n]);
A318672(n) = denominator(v318671_62[n]);
A318673(n) = valuation(A318672(n),2);

a(n) = denominator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d 1.

a(n) = 2^A318673(n).

A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -3

Offset: 1

Antti Karttunen, Sep 03 2018

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A049599, A318672 (denominators).

up_to = (2^16)+1;
DirSqrt(v) = {my(n=#v, u=vector(n)); u[1]=1; for(n=2, n, u[n]=(v[n]/v[1] - sumdiv(n, d, if(d>1&&dA049599(n) = factorback(apply(e -> (1+numdiv(e)),factor(n)[,2]));
v318671_72 = DirSqrt(vector(up_to, n, A049599(n)));
A318671(n) = numerator(v318671_72[n]);

a(n) = numerator of f(n), where f(1) = 1, f(n) = (1/2) * (A049599(n) - Sum_{d|n, d>1, d 1.

A004709 Cubefree numbers: numbers that are not divisible by any cube > 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85

Offset: 1

Steven Finch, Jun 14 1998

Numbers n such that no smaller number m satisfies: kronecker(n,k)=kronecker(m,k) for all k. - Michael Somos, Sep 22 2005
The asymptotic density of cubefree integers is the reciprocal of Apery's constant 1/zeta(3) = A088453. - Gerard P. Michon, May 06 2009
The Schnirelmann density of the cubefree numbers is 157/189 (Orr, 1969). - Amiram Eldar, Mar 12 2021
From Amiram Eldar, Feb 26 2024: (Start)
Numbers whose sets of unitary divisors (A077610) and bi-unitary divisors (A222266) coincide.
Number whose all divisors are (1+e)-divisors, or equivalently, numbers k such that A049599(k) = A000005(k). (End)

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Gérard P. Michon, On the number of cubefree integers not exceeding N.
Richard C. Orr, On the Schnirelmann density of the sequence of k-free integers, Journal of the London Mathematical Society, Vol. 1, No. 1 (1969), pp. 313-319.
Vladimir Shevelev, Set of all densities of exponentially S-numbers, arXiv preprint, arXiv:1511.03860 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Cubefree.

Complement of A046099.
Cf. A005117 (squarefree), A067259 (cubefree but not squarefree), A046099 (cubeful).
Cf. A160112, A160113, A160114 & A160115: On the number of cubefree integers. - Gerard P. Michon, May 06 2009
Cf. A030078.
Cf. A001221, A124010, A212793.
Cf. A000005, A049599, A077610, A222266, A376365, A376366.

a004709 n = a004709_list !! (n-1)
a004709_list = filter ((== 1) . a212793) [1..]
-- Reinhard Zumkeller, May 27 2012

isA004709 := proc(n)
    local p;
    for p in ifactors(n)[2] do
        if op(2,p) > 2 then
            return false;
        end if;
    end do:
    true ;
end proc:

Select[Range[6!], FreeQ[FactorInteger[#], {, k /; k > 2}] &] (* Jan Mangaldan, May 07 2014 *)

{a(n)= local(m,c); if(n<2, n==1, c=1; m=1; while( cvecmax(factor(m)[,2]), c++)); m)} /* Michael Somos, Sep 22 2005 */

from sympy.ntheory.factor_ import core
def ok(n): return core(n, 3) == n
print(list(filter(ok, range(1, 86)))) # Michael S. Branicky, Aug 16 2021

from sympy import mobius, integer_nthroot
def A004709(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**3) for k in range(1, integer_nthroot(x,3)[0]+1))
    m, k = n, f(n)
    while m != k:
        m, k = k, f(k)
    return m # Chai Wah Wu, Aug 05 2024

A066990(a(n)) = a(n). - Reinhard Zumkeller, Jun 25 2009
A212793(a(n)) = 1. - Reinhard Zumkeller, May 27 2012
A124010(a(n),k) <= 2 for all k = 1..A001221(a(n)). - Reinhard Zumkeller, Mar 04 2015
Sum_{n>=1} 1/a(n)^s = zeta(s)/zeta(3*s), for s > 1. - Amiram Eldar, Dec 27 2022

A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 1, 1, 3, 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, 3, 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, 3, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 1, 1

Offset: 1

Yasutoshi Kohmoto

The exponential divisors of a number x = Product p(i)^r(i) are all numbers of the form Product p(i)^s(i) where s(i) divides r(i) for all i.
Wu gives a complicated Dirichlet g.f.
a(1) = 1 by convention. This is also required for a function to be multiplicative. - N. J. A. Sloane, Mar 03 2009
The inverse Moebius transform seems to be in A124315. The Dirichlet inverse appears to be related to A166234. - R. J. Mathar, Jul 14 2014

			a(8)=2 because 2 and 2^3 are e-divisors of 8.
The sets of e-divisors start as:
  1:{1}
  2:{2}
  3:{3}
  4:{2, 4}
  5:{5}
  6:{6}
  7:{7}
  8:{2, 8}
  9:{3, 9}
  10:{10}
  11:{11}
  12:{6, 12}
  13:{13}
  14:{14}
  15:{15}
  16:{2, 4, 16}
  17:{17}
  18:{6, 18}
  19:{19}
  20:{10, 20}
  21:{21}
  22:{22}
  23:{23}
  24:{6, 24}

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Andrew V. Lelechenko, Exponential and infinitary divisors, arXiv:1405.7597 [math.NT], 2014, sequence tau^(e).
David Moews, A database of aliquot cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles.
J. O. M. Pedersen, Tables of Aliquot Cycles. [Via Internet Archive Wayback-Machine]
J. O. M. Pedersen, Tables of Aliquot Cycles. [Cached copy, pdf file only]
László Tóth and Nicuşor Minculete, Exponential unitary divisors, arXiv:0910.2798 [math.NT], 2009.
Tim Trudgian, The sum of the unitary divisor function, arXiv:1312.4615 [math.NT], 2013-2014, Section 3.
Eric Weisstein's World of Mathematics, e-Divisor.
Jie Wu, Problème de diviseurs exponentiels et entiers exponentiellement sans facteur carré, J. Theor. Nombr. Bordeaux 7 (1) (1995) 133-141.

Row lengths of A322791.
Cf. A049599, A061389, A051377 (sum of e-divisors).
Partial sums are in A099593.
Cf. A124010, A000005, A049599, A072911, A124315, A166234, A327837, A361012.

A049419:=n->Product(List(Collected(Factors(n)), p -> Tau(p[2]))); List([1..10^4], n -> A049419(n)); # Muniru A Asiru, Oct 29 2017

a049419 = product . map (a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A049419 := proc(n)
    local a;
    a := 1 ;
    for pf in ifactors(n)[2] do
        a := a*numtheory[tau](op(2,pf)) ;
    end do:
    a ;
end proc:
seq(A049419(n),n=1..20) ; # R. J. Mathar, Jul 14 2014

a[1] = 1; a[p_?PrimeQ] = 1; a[p_?PrimeQ, e_] := DivisorSigma[0, e]; a[n_] := Times @@ (a[#[[1]], #[[2]]] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 102}] (* Jean-François Alcover, Jan 30 2012, after Vladeta Jovovic *)

a(n) = vecprod(apply(numdiv, factor(n)[,2])); \\ Amiram Eldar, Mar 27 2023

Multiplicative with a(p^e) = tau(e). - Vladeta Jovovic, Jul 23 2001

Sum_{k=1..n} a(k) ~ A327837 * n. - Vaclav Kotesovec, Feb 27 2023

More terms from Jud McCranie, May 29 2000

A051378 Sum of (1+e)-divisors of n. Let n = Product_i p(i)^r(i) then (1+e)-sigma(n) = Product_i (1 + Sum_{s|r(i)} p(i)^s).

Original entry on oeis.org

1, 3, 4, 7, 6, 12, 8, 11, 13, 18, 12, 28, 14, 24, 24, 23, 18, 39, 20, 42, 32, 36, 24, 44, 31, 42, 31, 56, 30, 72, 32, 35, 48, 54, 48, 91, 38, 60, 56, 66, 42, 96, 44, 84, 78, 72, 48, 92, 57, 93, 72, 98, 54, 93, 72, 88, 80, 90, 60, 168, 62, 96, 104, 79, 84, 144, 68, 126, 96

Offset: 1

Yasutoshi Kohmoto

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A051377, A049599.

Cf. A027748, A124010, A027750, A069915, A107758, A107759.

a051378 n = product $ zipWith sum_1e (a027748_row n) (a124010_row n)
   where sum_1e p e = 1 + sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051378 := proc(n)
    local a,d,p,e,sp;
    a := 1;
    for d in ifactors(n)[2] do
        p := op(1,d) ;
        e := op(2,d) ;
        sp := 1;
        for s in numtheory[divisors](e) do
            sp := sp+p^s ;
        end do:
        a := a*sp ;
    end do:
    a;
end proc: # R. J. Mathar, Oct 26 2015

a[1] = 1; a[p_?PrimeQ] = p+1; a[n_] := Times @@ (1 + Sum[First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 69}] (* Jean-François Alcover, May 04 2012 *)

a(n)=my(f=factor(n));prod(i=1,#f[,1],sumdiv(f[i,2],d,f[i,1]^d)+1) \\ Charles R Greathouse IV, Nov 22 2011

Multiplicative with a(p^e) = 1 + Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = Sum_{d|n, gcd(d, n/d) = 1} A051377(d). - Daniel Suteu, Nov 01 2022
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + (1-1/p)*Sum_{k>=2} p^k/(p^(2*k)-1)) = 0.76636964336546210751... . - Amiram Eldar, Oct 31 2023

Corrected and extended by Naohiro Nomoto, Apr 12 2001

A072911 Number of "phi-divisors" of n.

Original entry on oeis.org

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

Offset: 1

Yasutoshi Kohmoto, Aug 21 2002

If n = Product p(i)^r(i), d = Product p(i)^s(i), = s(i)<=r(i) and gcd(s(i), r(i)) = 1, then d is a phi-divisor of n.

The integers n = Product_{i=1..r} p_i^{a_i} and m = Product_{i=1..r} p_i^{b_i}, a_i, b_i >= 1 (1 <= i <= r) having the same prime factors are called exponentially coprime, if gcd(a_i, b_i) = 1 for every 1 <= i <= r, i.e., the only common exponential divisor of n and m is Product_{i=1..r} p_i = the common squarefree kernel of n and m, cf. A049419, A007947. The terms of this sequence count the divisors d of n such that d and n are exponentially coprime. - Laszlo Toth, Oct 06 2008

József Sándor, On an exponential totient function, Studia Univ. Babes-Bolyai, Math., 41 (1996), 91-94. [Laszlo Toth, Oct 06 2008]

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, On certain arithmetic functions involving exponential divisors, Annales Univ. Sci. Budapest., Sect. Comp., 27 (2004), 285-294; arXiv:math/0610274 [math.NT], 2006-2009. [From _Laszlo Toth_, Oct 06 2008]

Cf. A061389, A327838.

Cf. A124010, A000010, A049599, A049419, A361012.

a072911 = product . map (a000010 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

A072911 := proc(n)
    local a, p;
    a := 1 ;
    for p in ifactors(n)[2] do
        a := a*numtheory[phi](op(2, p)) ;
    od:
    a ;
end:
seq(A072911(n),n=1..100) ; # R. J. Mathar, Sep 25 2008

a[n_] := Times @@ EulerPhi[FactorInteger[n][[All, 2]]];
Array[a, 105] (* Jean-François Alcover, Nov 16 2017 *)

If n = Product p(i)^r(i) then a(n) = Product (phi(r(i))), where phi(k) is the Euler totient function of k, cf. A000010.

Sum_{k=1..n} a(k) ~ c_1 * n + c_2 * n^(1/3) + O(n^(1/5+eps)), where c_1 = A327838 (Tóth, 2004). - Amiram Eldar, Oct 30 2022

More terms from R. J. Mathar, Sep 25 2008

A282446 Call d a recursive divisor of n iff the p-adic valuation of d is a recursive divisor of the p-adic valuation of n for any prime p dividing d; a(n) gives the number of recursive divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 3, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 5, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4

Offset: 1

Rémy Sigrist, Feb 15 2017

More informally, the prime tower factorization of a recursive divisor of n can be obtained by removing branches from the prime tower factorization of n (the prime tower factorization of a number is defined in A182318).
A recursive divisor of n is also a divisor of n, hence a(n)<=A000005(n) for any n, with equality iff n is cubefree (i.e. n belongs to A004709).
A recursive divisor of n is also a (1+e)-divisor of n, hence a(n)<=A049599(n) for any n, with equality iff the p-adic valuation of n is cubefree for any prime p dividing n.
This sequence first differs from A049599 at n=256: a(256)=4 whereas A049599(256)=5; note that 256=2^(2^3), and 2^3 is not cubefree.

			The recursive divisors of 40 are: 1, 2, 5, 8, 10 and 40, hence a(40)=6.

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Cf. A000005, A004709, A049599, A014221, A182318.

a[1] = 1; a[n_] := a[n] = Times @@ (1 + a/@ (Last /@ FactorInteger[n])); Array[a, 100] (* Amiram Eldar, Apr 12 2020 *)

a(n) = my (f=factor(n)); return (prod(i=1, #f~, 1+a(f[i,2])))

Multiplicative, with a(p^k)=1+a(k) for any prime p and k>0.

a(A014221(n))=n+1 for any n>=0.

A061389 Number of (1+phi)-divisors of n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 2, 3, 2, 4, 2, 4, 2, 4, 4, 3, 2, 4, 2, 4, 4, 4, 2, 6, 2, 4, 3, 4, 2, 8, 2, 5, 4, 4, 4, 4, 2, 4, 4, 6, 2, 8, 2, 4, 4, 4, 2, 6, 2, 4, 4, 4, 2, 6, 4, 6, 4, 4, 2, 8, 2, 4, 4, 3, 4, 8, 2, 4, 4, 8, 2, 6, 2, 4, 4, 4, 4, 8, 2, 6, 3, 4, 2, 8, 4, 4, 4, 6, 2, 8, 4, 4, 4, 4, 4, 10, 2, 4, 4, 4, 2, 8, 2, 6

Offset: 1

Vladeta Jovovic, Apr 29 2001

d is called a (1+phi)-divisor of a number n with prime factorization n = Product p(i)^r(i) if d|n and d = Product p(i)^s(i), where s(i)=0 or GCD(s(i),r(i))=1.

a(n) is odd iff n is a 3-full number (cf. A036966).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A069915, A049419, A049599.

Cf. A124010, A000010.

a061389 = product . map ((+ 1) . a000010 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

f[p_, e_] := EulerPhi[e] + 1; a[1] = 1; a[n_] := Times @@ ( f @@@ FactorInteger[n] ); Array[a, 100] (* Amiram Eldar, Aug 30 2019*)

Multiplicative with a(p^e) = A000010(e)+1.

A353898 a(n) is the number of divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, May 10 2022

First differs from A049599 and A282446 at n=32.

			The divisors of 8 are 1, 2 = 2^1, 4 = 2^2 and 8 = 2^3. 3 of these divisors, 1, 2 and 4, are in A138302. Therefore, a(8) = 3.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A004709, A049599, A138302, A282446, A353897.

Similar sequences: A034444, A048105, A286324, A049419.

f[p_, e_] := Floor[Log2[e]] + 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

Multiplicative with a(p^e) = floor(log_2(e)) + 2.
a(n) > 1 for n > 1 and a(n) = 2 if and only if n is a prime.
a(n) = A000005(n) if and only if n is cubefree (A004709).

A366901 The number of exponentially odious divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 3, 3, 4, 2, 6, 2, 4, 4, 4, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 3, 6, 2, 8, 2, 4, 4, 4, 4, 9, 2, 4, 4, 6, 2, 8, 2, 6, 6, 4, 2, 8, 3, 6, 4, 6, 2, 6, 4, 6, 4, 4, 2, 12, 2, 4, 6, 4, 4, 8, 2, 6, 4, 8, 2, 9, 2, 4, 6, 6, 4, 8, 2, 8, 4, 4, 2, 12, 4, 4, 4

Offset: 1

Amiram Eldar, Oct 27 2023

First differs from A049599 and A282446 at n = 32, from A365551 at n = 64, and from A353898 at n = 128.
The number of divisors of n that are exponentially odious numbers (A270428), i.e., numbers having only odious (A000069) exponents in their canonical prime factorization.
The sum of these divisors is A366903(n) and the largest of them is A366905(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000005, A000069, A004709, A115384, A270428, A366903, A366905.
Cf. A049599, A282446, A365551, A353898.
Similar sequences: A325837, A353898, A365680, A366902.

f[p_, e_] := Floor[e/2] + If[OddQ[e] || EvenQ[DigitCount[e + 1, 2, 1]], 1, 0] + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = 1 + n\2 + (n%2 || hammingweight(n+1)%2==0); \\ after Charles R Greathouse IV at A115384
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Multiplicative with a(p^e) = A115384(e) + 1.

a(n) <= A000005(n), with equality if and only if n is a cubefree number (A004709).

cp's OEIS Frontend

A318672 Denominators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

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A318671 Numerators of the sequence whose Dirichlet convolution with itself yields A049599, number of (1+e)-divisors of n.

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A004709 Cubefree numbers: numbers that are not divisible by any cube > 1.

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A049419 a(1) = 1; for n > 1, a(n) = number of exponential divisors of n.

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A051378 Sum of (1+e)-divisors of n. Let n = Product_i p(i)^r(i) then (1+e)-sigma(n) = Product_i (1 + Sum_{s|r(i)} p(i)^s).

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A072911 Number of "phi-divisors" of n.

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A282446 Call d a recursive divisor of n iff the p-adic valuation of d is a recursive divisor of the p-adic valuation of n for any prime p dividing d; a(n) gives the number of recursive divisors of n.

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