A124315 -id:A124315 - cp's OEIS frontend

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

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

A138010 a(n) is the number of positive divisors of n that divide d(n), where d(n) is the number of positive divisors of n, A000005(n); a(n) also equals d(gcd(n, d(n))).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 4, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 6, 1, 2, 2, 2, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 6, 1, 2, 2, 1, 1, 2, 1, 4, 1, 2, 1, 6, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4

Offset: 1

Leroy Quet, Feb 27 2008

nonn

			12 has 6 divisors (1,2,3,4,6,12). Those divisors of 12 that divide 6 are 1,2,3,6. Since there are 4 of these, then a(12) = 4.

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

Cf. A000005, A009191, A124315, A138011, A138012.

[#Divisors( Gcd(n,#Divisors(n))):n in [1..120]]; // Marius A. Burtea, Aug 03 2019

with(numtheory): a:=proc(n) local div,c,j: div:=divisors(n): c:=0: for j to tau(n) do if `mod`(tau(n), div[j])=0 then c:=c+1 else end if end do: c end proc: seq(a(n),n=1..90); # Emeric Deutsch, Mar 02 2008

Table[Length[Select[Divisors[n], Mod[Length[Divisors[n]], # ] == 0 &]], {n,1,100}] (* Stefan Steinerberger, Feb 29 2008 *)
Table[Count[DivisorSigma[0,n]/Divisors[n],?IntegerQ],{n,120}] (* _Harvey P. Dale, May 31 2019 *)

A138010(n) = sumdiv(n,d,if(!(numdiv(n)%d), 1, 0)); \\ Antti Karttunen, May 25 2017

from sympy import divisors, divisor_count
def a(n): return sum([ 1*(divisor_count(n)%d==0) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

(define (A138010 n) (A000005 (gcd n (A000005 n)))) ;; Antti Karttunen, May 25 2017

a(n) = A000005(A009191(n)). [From the alternative description.] - Antti Karttunen, May 25 2017

More terms from Stefan Steinerberger and Emeric Deutsch, Feb 29 2008

Further extended (to 120 terms) by Antti Karttunen, May 25 2017

A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).

Original entry on oeis.org

1, 3, 4, 8, 6, 12, 8, 18, 14, 18, 12, 32, 14, 24, 24, 39, 18, 42, 20, 48, 32, 36, 24, 72, 32, 42, 44, 64, 30, 72, 32, 81, 48, 54, 48, 112, 38, 60, 56, 108, 42, 96, 44, 96, 84, 72, 48, 156, 58, 96, 72, 112, 54, 132, 72, 144, 80, 90, 60, 192, 62, 96, 112, 166, 84

Offset: 1

Ilya Gutkovskiy, Feb 26 2020

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000005, A000010, A000203, A010052, A046951, A076752 (Mobius transf.), A124315, A206369, A344442, A347090 (Dirichlet inverse).

Table[Sum[Boole[IntegerQ[(n/d)^(1/2)]] DivisorSigma[1, d], {d, Divisors[n]}], {n, 1, 65}]
nmax = 65; CoefficientList[Series[Sum[DivisorSigma[1, k] (EllipticTheta[3, 0, x^k] - 1)/2, {k, 1, nmax}], {x, 0, nmax}], x] // Rest
f[p_, e_] := (2*p^(e + 3) - e*p^2 + e - If[OddQ[e], 3*p^2 - 1, 2*p^2 + 2*p - 2])/(2*(p - 1)*(p^2 - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* Amiram Eldar, May 25 2025 *)

A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d)); \\ Antti Karttunen, May 23 2021

G.f.: Sum_{k>=1} sigma(k) * (theta_3(x^k) - 1) / 2.
a(n) = Sum_{d|n} A076752(d).
a(n) = Sum_{d|n} A206369(n/d) * tau(d).
a(n) = Sum_{d|n} A010052(n/d) * sigma(d).
a(n) = Sum_{d|n} A124315(n/d) * phi(d).
a(n) = Sum_{d|n} A046951(n/d) * d.
a(p) = p + 1, where p is prime.
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1080. - Vaclav Kotesovec, Feb 26 2020
Multiplicative with a(p^e) = (2*p^(e+3) - e*p^2 + e - 3*p^2 + 1)/(2*(p-1)*(p^2-1)) if e is odd, and (2*p^(e+3) - e*p^2 + e - 2*p^2 - 2*p + 2)/(2*(p-1)*(p^2-1)) if e is even. - Amiram Eldar, May 25 2025

A124316 a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

Original entry on oeis.org

1, 2, 2, 5, 2, 4, 2, 8, 6, 4, 2, 10, 2, 4, 4, 15, 2, 12, 2, 10, 4, 4, 2, 16, 8, 4, 10, 10, 2, 8, 2, 22, 4, 4, 4, 30, 2, 4, 4, 16, 2, 8, 2, 10, 12, 4, 2, 30, 10, 16, 4, 10, 2, 20, 4, 16, 4, 4, 2, 20, 2, 4, 12, 37, 4, 8, 2, 10, 4, 8, 2, 48, 2, 4, 16, 10, 4, 8, 2, 30, 23, 4, 2, 20, 4, 4, 4, 16, 2, 24

Offset: 1

Robert G. Wilson v, Sep 30 2006

Apparently multiplicative and the inverse Mobius transform of A069290. - R. J. Mathar, Feb 07 2011

Antti Karttunen, Table of n, a(n) for n = 1..10000
Ekkehard Krätzel, Werner Georg Nowak, and László Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.
Manfred Kühleitner and Werner Georg Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions, Cent. Eur. J. Math., Vol. 11, No. 3 (2013), pp. 477-486; arXiv preprint, arXiv:1204.1146 [math.NT], 2012.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, in: T. Rassias, P. Pardalos (eds.) Mathematics Without Boundaries, Springer, New York, NY, 2024; arXiv preprint, arXiv:1310.7053 [math.NT], 2013-2014.

Cf. A000203, A001616, A001620, A069290, A061884, A082633, A124315.

A124316 := proc(n) local a,d;
  a := 0 ;
  for d in numtheory[divisors](n) do
     igcd(d,n/d) ;
     a := a+numtheory[sigma](%) ;
   end do:
   a;
end proc: # R. J. Mathar, Apr 14 2011

Table[Plus @@ Map[DivisorSigma[1, GCD[ #, n/# ]] &, Divisors@n], {n, 90}]
f[p_, e_] := (If[OddQ[e], 2*p^((e+3)/2), p^(e/2 + 1)*(p+1)] - (e+3)*p + e + 1)/(p-1)^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Mar 28 2024 *)

a(n) = sumdiv(n, d, sigma(gcd(d, n/d))); \\ Michel Marcus, Feb 13 2016

from sympy import divisors, divisor_sigma, gcd
def a(n): return sum([divisor_sigma(gcd(d, n/d)) for d in divisors(n)]) # Indranil Ghosh, May 25 2017

From Amiram Eldar, Mar 28 2024: (Start)
Multiplicative with a(p^e) = (p^(e/2 + 1)*(p+1) - (e+3)*p + e + 1)/(p-1)^2, if e is even, and (2*p^((e+3)/2) - (e+3)*p + e + 1)/(p-1)^2 if e is odd.
Dirichlet g.f.: zeta(s)^2 * zeta(2*s-1).
Sum_{k=1..n} a(k) = (log(n)^2/4 + (2*gamma - 1/2)*log(n) + 5*gamma^2/2 - 2*gamma - 3*gamma_1 + 1/2) * n + O(n^(2/3)*log(n)^(16/9)), where gamma is Euler's constant (A001620) and gamma_1 is the first Stieltjes constant (A082633) (Krätzel et al., 2012). (End)

A268732 Sum of the numbers of divisors of gcd(x,y) with x*y <= n.

Original entry on oeis.org

1, 3, 5, 9, 11, 15, 17, 23, 27, 31, 33, 41, 43, 47, 51, 60, 62, 70, 72, 80, 84, 88, 90, 102, 106, 110, 116, 124, 126, 134, 136, 148, 152, 156, 160, 176, 178, 182, 186, 198, 200, 208, 210, 218, 226, 230, 232, 250, 254, 262, 266, 274, 276, 288, 292, 304, 308, 312, 314, 330

Offset: 1

Michel Marcus, Feb 12 2016

nonn

Partial sums of A124315.

Daniel Suteu, Table of n, a(n) for n = 1..10000
Masum Billal, Asymptotic Result of A Generalization of A GCD-Sum, arXiv:2206.05023 [math.NT], 2022.
Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, arXiv:1602.03555 [math.HO], 2016. See Remark 1 p. 3.
Adrian W. Dudek, On the Success of Mishandling Euclid's Lemma, The American Mathematical Monthly, Vol. 123, No. 9 (2016), 924-927.
Randell Heyman, A summation involving the number of divisors function and the GCD function, arXiv:2003.13937 [math.NT], 2020.

Cf. A000005, A124315.

Cf. A001620, A013661, A306016.

Table[Total@ Flatten@ Map[Function[k, DivisorSigma[0, GCD[#, k]] & /@ Select[Range@ n, # k <= n &]], Range@ n], {n, 60}] (* Michael De Vlieger, Feb 12 2016 *)

a(n) = sum(k=1, n, sumdiv(k, d, numdiv(gcd(d, k/d))));

a(n) = sum(k=1, sqrtint(n), 2*sum(j=1, sqrtint(n\(k*k)), n\(j*k*k))-sqrtint(n\(k*k))^2); \\ Daniel Suteu, Jan 08 2019

a(n)=sum(k=1,n,sum(j=1,sqrt(n/k),floor(n/k/j^2))); \\ Benoit Cloitre, Oct 02 2022

a(n) = Sum_{k=1..floor(sqrt(n))} (2*Sum_{j=1..floor(sqrt(n/k^2))} floor(n/(j*k^2)) - floor(sqrt(n/k^2))^2). - Daniel Suteu, Jan 08 2019
a(n) = n*zeta(2)*(log(n) + 2*gamma - 1 + 2*zeta'(2)/zeta(2)) + O(sqrt(n)*log(n)), where gamma is the Euler-Mascheroni constant A001620. - Daniel Suteu, Jan 11 2019
a(n) = Sum_{i=1..n} Sum_{j=1..n} floor(sqrt(n/(i*j))). - Ridouane Oudra, Apr 13 2025

A344442 a(n) = A332844(n) - n.

Original entry on oeis.org

0, 1, 1, 4, 1, 6, 1, 10, 5, 8, 1, 20, 1, 10, 9, 23, 1, 24, 1, 28, 11, 14, 1, 48, 7, 16, 17, 36, 1, 42, 1, 49, 15, 20, 13, 76, 1, 22, 17, 68, 1, 54, 1, 52, 39, 26, 1, 108, 9, 46, 21, 60, 1, 78, 17, 88, 23, 32, 1, 132, 1, 34, 49, 102, 19, 78, 1, 76, 27, 74, 1, 180, 1, 40, 53, 84, 19, 90, 1, 154, 54, 44, 1, 172, 23, 46

Offset: 1

Antti Karttunen, May 23 2021

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10080
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000010, A124315, A332844.

Table[DivisorSum[n, Boole[IntegerQ[(n/#)^(1/2)]] DivisorSigma[1, #] &] - n, {n, 86}] (* Michael De Vlieger, May 24 2021 *)

A332844(n) = sumdiv(n,d, issquare(n/d) * sigma(d));
A344442(n) = (A332844(n) - n);

a(n) = A332844(n) - n.
a(n) = Sum_{d|n} A000010(n/d) * (A124315(d)-1).
Sum_{k=1..n} a(k) ~ c * n^2 /  2, where c = Pi^6/540 - 1 = 0.7803503... . - Amiram Eldar, Dec 04 2024

A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.

Original entry on oeis.org

1, 3, 3, 5, 3, 9, 3, 8, 5, 9, 3, 15, 3, 9, 9, 11, 3, 15, 3, 15, 9, 9, 3, 24, 5, 9, 8, 15, 3, 27, 3, 15, 9, 9, 9, 25, 3, 9, 9, 24, 3, 27, 3, 15, 15, 9, 3, 33, 5, 15, 9, 15, 3, 24, 9, 24, 9, 9, 3, 45, 3, 9, 15, 19, 9, 27, 3, 15, 9, 27, 3, 40, 3, 9, 15

Offset: 1

Ilya Gutkovskiy, Feb 21 2020

Inverse Moebius transform of A322483.

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

Cf. A000005, A007425, A024206, A048691, A124315, A124316, A295316, A322483, A332619, A332713.

Table[Sum[DivisorSigma[0, d/GCD[d, n/d]], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e+3)/2]; A322483[n_] := If[n==1, 1, Times @@ (f @@@ FactorInteger[n])]; Table[Sum[A322483[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Floor[(e + 1)*(e + 5)/4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Dec 05 2022 *)

a(n) = Sum_{d|n} A322483(d).
a(n) = Sum_{d|n} tau(n/d) * A295316(d).
Multiplicative with a(p^e) = floor((e+1)*(e+5)/4) = A024206(e+2). - Amiram Eldar, Dec 05 2022

cp's OEIS Frontend

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

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A138010 a(n) is the number of positive divisors of n that divide d(n), where d(n) is the number of positive divisors of n, A000005(n); a(n) also equals d(gcd(n, d(n))).

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A332844 Dirichlet g.f.: zeta(s) * zeta(s-1) * zeta(2*s).

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A124316 a(n) = Sum_{d|n} sigma(gcd(d,n/d)), where sigma is the sum of divisors function, A000203.

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A268732 Sum of the numbers of divisors of gcd(x,y) with x*y <= n.

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A344442 a(n) = A332844(n) - n.

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A332730 a(n) = Sum_{d|n} tau(d/gcd(d, n/d)), where tau = A000005.