A322791 -id:A322791 - cp's OEIS frontend

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

1, 2, 3, 6, 5, 6, 7, 10, 12, 10, 11, 18, 13, 14, 15, 22, 17, 24, 19, 30, 21, 22, 23, 30, 30, 26, 30, 42, 29, 30, 31, 34, 33, 34, 35, 72, 37, 38, 39, 50, 41, 42, 43, 66, 60, 46, 47, 66, 56, 60, 51, 78, 53, 60, 55, 70, 57, 58, 59, 90, 61, 62, 84, 78, 65, 66, 67, 102, 69, 70, 71

Offset: 1

Yasutoshi Kohmoto

The e-divisors (or exponential divisors) of 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.

a(n) = n if and only if n is squarefree. - Jon Perry, Nov 13 2012

			a(8)=10 because 2 and 2^3 are e-divisors of 8 and 2+2^3=10.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
J. Fabrykowski and M. V. Subbarao, The maximal order and the average order of multiplicative function sigma^(e)(n), in Théorie des nombres/Number theory (Quebec, PQ, 1987), 201-206, de Gruyter, Berlin, 1989.
Nicussor Minculete, Concerning some arithmetic functions which use exponential divisors, Acta Universitatis Apulensis, No. 28/2011, pp. 125-133.
Y.-F. S. Pétermann and J. Wu, On the sum of exponential divisors of an integer, Acta Math. Hungar. 77 (1997), 159-175.
Eric Weisstein's World of Mathematics, e-Divisor

Cf. A051378, A049419 (number of e-divisors).
Cf. A027748, A124010, A027750.
Row sums of A322791.
See A307042 and A275480 where the formula and constant appear.

A051377:=n->Product(List(Collected(Factors(n)), p -> Sum(DivisorsInt(p[2]),d->p[1]^d))); List([1..10^4], n -> A051377(n)); # Muniru A Asiru, Oct 29 2017

a051377 n = product $ zipWith sum_e (a027748_row n) (a124010_row n) where
   sum_e p e = sum [p ^ d | d <- a027750_row e]
-- Reinhard Zumkeller, Mar 13 2012

A051377 := proc(n)
    local a,pe,p,e;
    a := 1;
    for pe in ifactors(n)[2] do
        p := pe[1] ;
        e := pe[2] ;
        add(p^d,d=numtheory[divisors](e)) ;
        a := a*% ;
    end do:
    a ;
end proc:
seq(A051377(n),n=1..100) ; # R. J. Mathar, Oct 05 2017

a[n_] := Times @@ (Sum[ First[#]^d, {d, Divisors[Last[#]]}] & ) /@ FactorInteger[n]; Table[a[n], {n, 1, 71}] (* Jean-François Alcover, Apr 06 2012 *)

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

ediv(n,f=factor(n))=my(v=List(),D=apply(divisors,f[,2]~),t=#f~); forvec(u=vector(t,i,[1,#D[i]]), listput(v,prod(j=1,t,f[j,1]^D[j][u[j]]))); Set(v)
a(n)=vecsum(ediv(n)) \\ Charles R Greathouse IV, Oct 29 2018

Multiplicative with a(p^e) = Sum_{d|e} p^d. - Vladeta Jovovic, Apr 23 2002
a(n) = A126164(n)+n. - R. J. Mathar, Oct 05 2017
The average order of a(n) is Dn + O(n^e) for any e > 0, due to Fabrykowski & Subbarao, where D is about 0.568. (D >= 0.5 since a(n) >= n.) - Charles R Greathouse IV, Sep 22 2023

More terms from Jud McCranie, May 29 2000

Definition corrected by Jaroslav Krizek, Feb 27 2009

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

A056624 Number of unitary square divisors of n.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Aug 08 2000

Unitary analog of A046951.

The number of exponential divisors (A322791) of n that are cubefree (A004709). - Amiram Eldar, Jun 03 2025

			n=256, it has 5 square divisors of which only 2,{1,256} are unitary, 3 divisors are not.
n=124 has 2 (1 and 4) square divisors, both of them unitary a(124) = 2.
n=108 has 12 divisors, 4 square divisors: {1,4,9,36} of which 1 and 4 are unitary, 9 and 36 are not. So a(108)=2. The largest unitary square divisor of 108 is 4 with 1 prime divisor so a(108) = 2^1 = 2.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A000188, A004709, A008833, A034444, A046952, A055229, A056626, A059841, A162641, A322791, A350388.

isA056624 := (n, d) -> igcd(n, d) = d and igcd(n/d, d) = d and igcd(n/d^2, d) = 1:
a := n -> nops(select(k -> isA056624(n, k), [seq(1..n)])):  # Peter Luschny, Jun 13 2025

Table[DivisorSum[n, 1 &, And[IntegerQ@ Sqrt@ #, CoprimeQ[#, n/#]] &], {n, 105}] (* Michael De Vlieger, Jul 28 2017 *)
f[p_, e_] := 2^(1 - Mod[e, 2]); a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 03 2022 *)

a(n) = sumdiv(n, d, if(gcd(d, n/d)==1, issquare(d))); \\ Michel Marcus, Jul 28 2017

from sympy import factorint
def A056624(n): return 1<Chai Wah Wu, Aug 03 2024

def is_A056624(n, d): return gcd(n, d) == d and gcd(n//d, d) == d and gcd(n//(d*d), d) == 1
def a(n): return len([k for k in range(1, n+1) if is_A056624(n, k)])
print([a(n) for n in range(1, 106)])  # Peter Luschny, Jun 13 2025

(define (A056624 n) (if (= 1 n) n (* (A000079 (A059841 (A067029 n))) (A056624 (A028234 n))))) ;; Antti Karttunen, Jul 28 2017

a(n) = 2^r, where r is the number of prime factors of the largest unitary square divisor of n.
Multiplicative with a(p^e) = 2^(1-(e mod 2)). - Vladeta Jovovic, Dec 13 2002
Dirichlet g.f.: zeta(s)*zeta(2*s)/zeta(3*s). - Werner Schulte, Apr 03 2018
Sum_{k=1..n} a(k) ~ n*Pi^2/(6*zeta(3)) + sqrt(n)*zeta(1/2)/zeta(3/2). - Vaclav Kotesovec, Feb 07 2019
a(n) = 2^A162641(n). - Amiram Eldar, Sep 26 2022
a(n) = A034444(A350388(n)). - Amiram Eldar, Sep 09 2023

More terms from Vladeta Jovovic, Dec 13 2002

A049599 Number of (1+e)-divisors of n: if n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor 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, 4, 6, 2, 12, 4, 6, 4, 4, 4, 6, 2, 6, 6, 9, 2, 8, 2

Offset: 1

Yasutoshi Kohmoto

A divisor of n is a (1+e)-divisor if and only if it is a unitary divisor of an exponential divisor of n (see A077610 and A322791). - Amiram Eldar, Feb 26 2024

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

Cf. A049603, A051378.

Cf. A124010, A000005, A049419, A077610, A322791.

a049599 = product . map ((+ 1) . a000005 . fromIntegral) . a124010_row
-- Reinhard Zumkeller, Mar 13 2012

a[n_] := Times @@ (DivisorSigma[0, #] + 1 &)  /@ FactorInteger[n][[All, 2]]; a[1] = 1; Table[a[n], {n, 1, 103}] (* Jean-François Alcover, Oct 10 2011 *)

a(n) = vecprod(apply(x->numdiv(x)+1, factor(n)[, 2])); \\ Amiram Eldar, Aug 13 2023

If n = Product p(i)^r(i) then a(n) = Product (tau(r(i))+1), where tau(n) = number of divisors of n, cf. A000005. - Vladeta Jovovic, Apr 29 2001

More terms from Naohiro Nomoto, Apr 12 2001

A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

Original entry on oeis.org

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, 2, 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, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Dec 30 2022

First differs from A318672 and A325989 at n = 32.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both an infinitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e and the bitwise AND of e and k is equal to k.
The least term that is higher than 2 is a(216) = 4.
The position of the first appearance of a prime p in this sequence is 2^A359081(p), if A359081(p) > -1. E.g., 2^39 = 549755813888 for p = 3, 2^175 = 4.789...*10^52 for p = 5, and 2^1275 = 6.504...*10^383 for p = 7.
This sequence is unbounded since A246600 is unbounded (see A359082).

			a(8) = 2 since 8 has 2 divisors that are both infinitary and exponential: 2 and 8.

Cf. A037445, A049419, A077609, A080948, A138302, A246600, A322791, A359081, A359082, A359412.

Cf. A318672, A325989.

s[n_] := DivisorSum[n, 1 &, BitAnd[n, #] == # &]; f[p_, e_] := s[e]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

s(n) = sumdiv(n, d, bitand(d, n)==d);
a(n) = {my(f = factor(n)); prod(i = 1, #f~, s(f[i,2]));}

from math import prod
from sympy import divisors, factorint
def A359411(n): return prod(sum(1 for d in divisors(e,generator=True) if e|d == e) for e in factorint(n).values()) # Chai Wah Wu, Sep 01 2023

Multiplicative with a(p^e) = A246600(e).
a(n) = 1 if and only if n is in A138302.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} A246600(k)/p^k) = 1.135514937... .

A348964 Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 21, 22, 23, 26, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 46, 47, 51, 53, 55, 57, 58, 59, 60, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 75, 77, 78, 79, 82, 83, 84, 85, 86, 87, 89, 90, 91, 93, 94

Offset: 1

Amiram Eldar, Nov 05 2021

nonn

Sándor (2006) proved that all the squarefree numbers are e-harmonic of type 2.
Equivalently, numbers k such that A348963(k) | k * A049419(k).
Apparently, most exponential harmonic numbers of type 1 (A348961) are also terms of this sequence. Those that are not exponential harmonic numbers of type 2 are 1936, 5808, 9680, 13552, 17424, 29040, ...

			The squarefree numbers are trivial terms. If k is squarefree, then it has a single exponential divisor, k itself, and thus the harmonic mean of its exponential divisors is also k, which is an integer.
12 is a term since its exponential divisors are 6 and 12, and their harmonic mean, 8, is an integer.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Nicuşor Minculete, Contribuţii la studiul proprietăţilor analitice ale funcţiilor aritmetice - Utilizarea e-divizorilor, Ph.D. thesis, Academia Română, 2012. See section 4.3, pp. 90-94.
József Sándor, On exponentially harmonic numbers, Scientia Magna, Vol. 2, No. 3 (2006), pp. 44-47.
József Sándor, Selected Chapters of Geomety, Analysis and Number Theory, 2005, pp. 141-145.

A005117 and A348965 are subsequences.
Cf. A049419, A322791, A348961, A348963.
Similar sequences: A001599, A006086, A063947, A286325, A319745.

f[p_, e_] := p^e * DivisorSigma[0, e] / DivisorSum[e, p^(e-#) &]; ehQ[1] = True; ehQ[n_] := IntegerQ[Times @@ f @@@ FactorInteger[n]]; Select[Range[100], ehQ]

A361255 Triangle read by rows: row n lists the exponential unitary divisors of n.

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 6, 7, 2, 8, 3, 9, 10, 11, 6, 12, 13, 14, 15, 2, 16, 17, 6, 18, 19, 10, 20, 21, 22, 23, 6, 24, 5, 25, 26, 3, 27, 14, 28, 29, 30, 31, 2, 32, 33, 34, 35, 6, 12, 18, 36, 37, 38, 39, 10, 40, 41, 42, 43, 22, 44, 15, 45, 46, 47, 6, 48, 7, 49, 10, 50, 51, 26, 52, 53, 6, 54, 55, 14, 56, 57, 58, 59

Offset: 1

R. J. Mathar, Mar 06 2023

Starts to differ from A322791 in row n=16, where 4 is an exponential divisor but not an exponential unitary divisor.

Amiram Eldar, Table of n, a(n) for n = 1..15331 (first 10000 rows, flattened)
Nicuşor Minculete and László Tóth, Exponential unitary divisors, Annales Univ. Sci. Budapest., Sect. Comp., Vol. 35 (2011), pp. 205-216.

Cf. A322857 (row sums), A278908 (row lengths), A322791 (includes non-unitary exp divs).

A361255 := proc(n)
    local expundivs ,d,isue,p,ai,bi;
    expudvs := {} ;
    for d in numtheory[divisors](n) do
        isue := true ;
        for p in numtheory[factorset](n) do
            ai := padic[ordp](n,p) ;
            bi := padic[ordp](d,p) ;
            if bi > 0 then
                if modp(ai,bi) <>0 or igcd(bi,ai/bi) <> 1 then
                    isue := false;
                end if;
            else
                isue := false ;
            end if;
        end do;
        if isue then
            expudvs := expudvs union {d} ;
        end if;
    end do:
    sort(expudvs) ;
end proc:
seq(op(A361255(n)),n=1..60) ;

udivQ[n_, m_] := (n > 0 && m > 0 && Divisible[n, m] && CoprimeQ[m, n/m]);
expuDivQ[n_, d_] := Module[{f = FactorInteger[n]}, And @@ MapThread[udivQ, {f[[;; , 2]], IntegerExponent[d, f[[;; , 1]]]}]]; expuDivs[1] = {1};
expuDivs[n_] := Module[{d = Rest[Divisors[n]]}, Select[d, expuDivQ[n, #] &]];
Table[expuDivs[n], {n, 1, 70}] // Flatten (* Amiram Eldar, Mar 11 2023 *)

A368979 The number of exponential divisors of n that are exponentially odd numbers (A268335).

Original entry on oeis.org

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, 2, 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, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jan 11 2024

First differs from A367516 at n = 128, and from A359411 at n = 512.

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

Cf. A001227, A008578, A049419, A138302, A268335, A322791, A368980.
Cf. A359411, A367516.
Similar sequences: A055076, A322483, A363825, A368977.

f[p_, e_] := DivisorSigma[0, e/2^IntegerExponent[e, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = vecprod(apply(x -> numdiv(x >> valuation(x, 2)), factor(n)[, 2]));

Multiplicative with a(p^e) = A001227(e).
a(n) >= 1, with equality if and only if n is in A138302.
a(n) <= A049419(n), with equality if and only if n is noncomposite (A008578).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=2} (d(k) - d(k-1))/p^k) = 1.13657098749361390865..., where d(k) is the number of odd divisors of k (A001227).

A362852 The number of divisors of n that are both bi-unitary and exponential.

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

Offset: 1

Amiram Eldar, May 05 2023

First differs from A061704 at n = 128, and from A304327 and abs(A307428) at n = 64.
If e > 0 is the exponent of the highest power of p dividing n (where p is a prime), then for each divisor d of n that is both a bi-unitary and an exponential divisor, the exponent of the highest power of p dividing d is a number k such that k | e but k != e/2.
The least term that is higher than 2 is a(64) = 3.
This sequence is unbounded. E.g., a(2^(2^prime(n))) = prime(n).

			a(8) = 2 since 8 has 2 divisors that are both bi-unitary and exponential: 2 and 8.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Biunitary Divisor.
Eric Weisstein's World of Mathematics, e-Divisor.

Cf. A000005, A004709, A049419, A188999, A222266, A286324, A322791, A359411, A362853 (indices of records), A362854.

Cf. A061704, A304327, A307428.

f[p_, e_] := DivisorSigma[0, e] - If[OddQ[e], 0, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, numdiv(f[i, 2]) - !(f[i, 2] % 2));}

Multiplicative with a(p^e) = d(e) if e is odd, and d(e)-1 if e is even, where d(k) is the number of divisors of k (A000005).
a(n) = 1 if and only if n is cubefree (A004709).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} (1 + Sum_{k>=1} (d(k)+(k mod 2)-1)/p^k) = 1.1951330849... .

A331737 Multiplicative with a(p^e) = p^A000265(e), where A000265(x) gives the odd part of x.

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 8, 3, 10, 11, 6, 13, 14, 15, 2, 17, 6, 19, 10, 21, 22, 23, 24, 5, 26, 27, 14, 29, 30, 31, 32, 33, 34, 35, 6, 37, 38, 39, 40, 41, 42, 43, 22, 15, 46, 47, 6, 7, 10, 51, 26, 53, 54, 55, 56, 57, 58, 59, 30, 61, 62, 21, 8, 65, 66, 67, 34, 69, 70, 71, 24, 73, 74, 15, 38, 77, 78, 79, 10, 3, 82, 83, 42, 85, 86, 87, 88, 89, 30

Offset: 1

Antti Karttunen, Feb 02 2020

a(n) is the largest exponential divisor of n (cf. A322791) that is an exponentially odd number (A268335). - Amiram Eldar, Nov 17 2022

Antti Karttunen, Table of n, a(n) for n = 1..16384
Brahim Mittou, New properties of an arithmetic function, Mathematica Montisnigri, Vol LIII (2022).

Cf. A000265, A268335, A322791, A331738.

f[p_, e_] := p^(e/2^IntegerExponent[e, 2]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

A000265(n) = (n>>valuation(n,2));
A331737(n) = { my(f = factor(n)); prod(k=1, #f~, f[k, 1]^A000265(f[k, 2])); };

a(n) = n / A331738(n).

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((1-1/p) * Sum_{k>=1} p^(2^k - 1)/(p^(2^(k+1)-2) - 1)) = 0.3953728204... . - Amiram Eldar, Nov 17 2022

cp's OEIS Frontend

A051377 a(1)=1; for n > 1, a(n) = sum of exponential divisors (or e-divisors) of n.

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

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A056624 Number of unitary square divisors of n.

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A049599 Number of (1+e)-divisors of n: if n = Product p(i)^r(i), d = Product p(i)^s(i) and s(i) = 0 or s(i) divides r(i), then d is a (1+e)-divisor of n.

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A359411 a(n) is the number of divisors of n that are both infinitary and exponential.

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A348964 Exponential harmonic (or e-harmonic) numbers of type 2: numbers k such that the harmonic mean of the exponential divisors of k is an integer.

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