keyword:mult - cp's OEIS frontend

A005361 Product of exponents of prime factorization of n.

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 2, 1, 1, 1, 3, 2, 1, 3, 2, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 2, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 2, 1, 1, 2, 6, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1

Offset: 1

Jeffrey Shallit and Olivier Gérard

a(n) depends only on prime signature of n (cf. A025487, A052306). So a(24) = a(375) since 24 = 2^3*3 and 375 = 3*5^3 both have prime signature (3,1).
There was a comment here that said "a(n) is the number of nilpotents elements in the ring Z/nZ", but this is false, see A003557.
a(n) is the number of square-full divisors of n. a(n) is also the number of divisors d of n such that d and n have the same prime factors, i.e., A007947(d) = A007947(n). - Laszlo Toth, May 22 2009
Number of divisors u of n such that u|(u^n/n). Row lengths in triangle of A284318. - Juri-Stepan Gerasimov, Apr 05 2017

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel Forgues, Table of n, a(n) for n = 1..100000 (first 10000 terms from T. D. Noe)
Imanuel Chen and Michael Z. Spivey, Integral Generalized Binomial Coefficients of Multiplicative Functions, Preprint 2015; Summer Research Paper 238, Univ. Puget Sound.
P. Erdős and T. Motzkin, Problem 5735, Amer. Math. Monthly, 78 (1971), 680-681. (Incorrect solution!)
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
J. Knopfmacher, A prime-divisor function, Proc. Amer. Math. Soc., 40 (1973), 373-377.
MathStackExchange, Dirichlet series for zeta(s)*zeta(2*s)*zeta(3*s)*zeta(6*s)^(-1).
H. N. Shapiro, Problem 5735, Amer. Math. Monthly, 97 (1990), 937.
D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc., Vol. 34, No. 1 (1972), pp. 79-80.
Index entries for sequences computed from exponents in factorization of n

Cf. A000005, A002110, A005117 (indices of ones), A028234, A052306, A067029, A072411, A082695, A284318.
Cf. A360969, A360970, A361132.
Cf. A340065 (Dgf at s=2).

a005361 = product . a124010_row -- Reinhard Zumkeller, Jan 09 2012

A005361 := proc(n)
    local a, p ;
    a := 1 ;
    for p in ifactors(n)[2] do
       a := a*op(2, p) ;
    end do:
    a ;
end proc:
seq(A005361(n),n=1..30) ; # R. J. Mathar, Nov 20 2012
# second Maple program:
a:= n-> mul(i[2], i=ifactors(n)[2]):
seq(a(n), n=1..80);  # Alois P. Heinz, Feb 18 2020

Prepend[ Array[ Times @@ Last[ Transpose[ FactorInteger[ # ] ] ]&, 100, 2 ], 1 ]
Array[Times@@Transpose[FactorInteger[#]][[2]]&,80] (* Harvey P. Dale, Aug 15 2012 *)

for(n=1,100, f=factor(n); print1(prod(i=1,omega(f), f[i,2]),",")) \\ edited by M. F. Hasler, Feb 18 2020

a(n)=factorback(factor(n)[,2]) \\ Charles R Greathouse IV, Nov 07 2014

for(n=1, 100, print1(direuler(p=2, n, (1 - X + X^2)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 14 2020

from math import prod
from sympy import factorint
def a(n): return prod(factorint(n).values())
print([a(n) for n in range(1, 91)]) # Michael S. Branicky, Jul 04 2022

(define (A005361 n) (if (= 1 n) 1 (* (A067029 n) (A005361 (A028234 n))))) ;; Antti Karttunen, Mar 06 2017

n = Product (p_j^k_j) -> a(n) = Product (k_j).
Dirichlet g.f.: zeta(s)*zeta(2s)*zeta(3s)/zeta(6s).
Multiplicative with a(p^e) = e. - David W. Wilson, Aug 01 2001
a(n) = Sum_{d dividing n} floor(rad(d)/rad(n)) where rad(n) is A007947. - Enrique Pérez Herrero, Nov 06 2009
For n > 1: a(n) = Product_{k=1..A001221(n)} A124010(n,k). - Reinhard Zumkeller, Aug 27 2011
a(n) = tau(n/rad(n)), where tau is A000005 and rad is A007947. - Anthony Browne, May 11 2016
a(n) = Sum_{k=1..n}(floor(cos^2(Pi*k^n/n))*floor(cos^2(Pi*n/k))). - Anthony Browne, May 11 2016
From Antti Karttunen, Mar 06 2017: (Start)
For all n >= 1, a(prime^n) = n, a(A002110(n)) = a(A005117(n)) = 1. [From Crossrefs section.]
a(1) = 1; for n > 1, a(n) = A067029(n) * a(A028234(n)).
(End)
Let (b(n)) be multiplicative with b(p^e) = -1 + ( (floor((e-1)/3)+floor(e/3)) mod 4 ) for p prime and e > 0, then b(n) is the Dirichlet inverse of (a(n)). - Werner Schulte, Feb 23 2018
Sum_{i=1..k} a(i) ~ (zeta(2)*zeta(3)/zeta(6)) * k (Suryanarayana and Sitaramachandra Rao, 1972). - Amiram Eldar, Apr 13 2020
More precise asymptotics: Sum_{k=1..n} a(k) ~ 315*zeta(3)*n / (2*Pi^4) + zeta(1/2)*zeta(3/2)*sqrt(n) / zeta(3) + 6*zeta(1/3)*zeta(2/3)*n^(1/3) / Pi^2 [Knopfmacher, 1973]. - Vaclav Kotesovec, Jun 13 2020

A047994 Unitary totient (or unitary phi) function uphi(n).

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 7, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 14, 24, 12, 26, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 28, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 26, 40, 42, 36, 28, 58, 24, 60, 30, 48, 63, 48, 20, 66, 48, 44, 24, 70

Offset: 1

N. J. A. Sloane

A divisor d of n is called a unitary divisor if gcd(d, n/d) = 1. Define gcd*(k,n) to be the largest divisor d of k that is also a unitary divisor of n (that is, such that gcd(d, n/d) = 1). The unitary totient function a(n) = number of k with 1 <= k <= n such that gcd*(k,n) = 1. - N. J. A. Sloane, Aug 08 2021
Unitary convolution of A076479 and A000027. - R. J. Mathar, Apr 13 2011
Multiplicative with a(p^e) = p^e - 1. - N. J. A. Sloane, Apr 30 2013

			a(12) = a(3)*a(4) = 2*3 = 6.

T. D. Noe, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr. 74 (1960) 66-80
Steven R. Finch, Unitarism and infinitarism.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
M. Lal, Iterates of the unitary totient function, Math. Comp., 28 (1974), 301-302.
R. J. Mathar, Survey of Dirichlet Series of Multiplicative Arithmetic Functions, arXiv:1106.4038 [math.NT], 2011, Remark 43.
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009), Article 09.5.2.
László Tóth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010), Article 10.8.1.

Cf. A000010, A000203, A001221, A003557, A049865, A003271, A028233, A076479.

a047994 n = f n 1 where
   f 1 uph = uph
   f x uph = f (x `div` sppf) (uph * (sppf - 1)) where sppf = a028233 x
-- Reinhard Zumkeller, Aug 17 2011

A047994 := proc(n)
    local a, f;
    a := 1 ;
    for f in ifactors(n)[2] do
        a := a*(op(1,f)^op(2,f)-1) ;
    end do:
    a ;
end proc:
seq(A047994(n),n=1..20) ; # R. J. Mathar, Dec 22 2011

uphi[n_] := (Times @@ (Table[ #[[1]]^ #[[2]] - 1, {1} ] & /@ FactorInteger[n]))[[1]]; Table[ uphi[n], {n, 2, 75}] (* Robert G. Wilson v, Sep 06 2004 *)
uphi[n_] := If[n==1, 1, Product[{p, e} = pe; p^e-1, {pe, FactorInteger[n]}] ]; Array[uphi, 80] (* Jean-François Alcover, Nov 17 2018 *)

A047994(n)=my(f=factor(n)~); prod(i=1, #f, f[1, i]^f[2, i]-1);

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

from math import prod
from sympy import factorint
def A047994(n): return prod(p**e-1 for p, e in factorint(n).items()) # Chai Wah Wu, Sep 24 2021

If n = Product p_i^e_i, uphi(n) = Product (p_i^e_i - 1).
a(n) = A000010(n)*A000203(A003557(n))/A003557(n). - Velin Yanev and Charles R Greathouse IV, Aug 23 2017
From Amiram Eldar, May 29 2020: (Start)
a(n) = Sum_{d|n, gcd(d, n/d) = 1} (-1)^omega(d) * n/d.
Sum_{d|n, gcd(d, n/d) = 1} a(d) = n.
a(n) >= phi(n) = A000010(n), with equality if and only if n is squarefree (A005117). (End)
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^2 / 12, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3). - Vaclav Kotesovec, Jun 15 2020
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 2/p^s + 1/p^(2*s-1)). - Amiram Eldar, May 22 2025

More terms from Jud McCranie

A023900 Dirichlet inverse of Euler totient function (A000010).

Original entry on oeis.org

1, -1, -2, -1, -4, 2, -6, -1, -2, 4, -10, 2, -12, 6, 8, -1, -16, 2, -18, 4, 12, 10, -22, 2, -4, 12, -2, 6, -28, -8, -30, -1, 20, 16, 24, 2, -36, 18, 24, 4, -40, -12, -42, 10, 8, 22, -46, 2, -6, 4, 32, 12, -52, 2, 40, 6, 36, 28, -58, -8, -60, 30, 12, -1, 48, -20, -66, 16, 44, -24, -70, 2, -72, 36, 8, 18, 60, -24, -78, 4, -2

Offset: 1

Olivier Gérard

Also called reciprocity balance of n.
Apart from different signs, same as Sum_{d divides n} core(d)*mu(n/d), where core(d) (A007913) is the squarefree part of d. - Benoit Cloitre, Apr 06 2002
Main diagonal of A191898. - Mats Granvik, Jun 19 2011

			x - x^2 - 2*x^3 - x^4 - 4*x^5 + 2*x^6 - 6*x^7 - x^8 - 2*x^9 + 4*x^10 - ...

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 37.
D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc. Boston, MA, 1976, p. 125.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)
G. P. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
K. Dohmen and M. Trinks, An Abstraction of Whitney's Broken Circuit Theorem, arXiv preprint arXiv:1404.5480 [math.CO], 2014.
R. Kemp, On the number of words in the language {w in Sigma* | w = w^R }^2, Discrete Math., 40 (1982), 225-234.
Paul W. Oxby, A Function Based on Chebyshev Polynomials as an Alternative to the Sinc Function in FIR Filter Design, arXiv:2011.10546 [eess.SP], 2020.
László Tóth, Multiplicative arithmetic functions of several variables: a survey, arXiv preprint arXiv:1310.7053 [math.NT], 2013.

Cf. A000010, A007913, A023898, A117209, A134842.
Moebius transform is A055615.
Cf. A027748, A173557 (gives the absolute values), A295876.
Cf. A253905 (Dgf at s=3).

a023900 1 = 1
a023900 n = product $ map (1 -) $ a027748_row n
-- Reinhard Zumkeller, Jun 01 2015

A023900 := n -> mul(1-i,i=numtheory[factorset](n)); # Peter Luschny, Oct 26 2010

a[ n_] := If[ n < 1, 0, Sum[ d MoebiusMu @ d, { d, Divisors[n]}]] (* Michael Somos, Jul 18 2011 *)
Array[ Function[ n, 1/Plus @@ Map[ #*MoebiusMu[ # ]/EulerPhi[ # ]&, Divisors[ n ] ] ], 90 ]
nmax = 81; Drop[ CoefficientList[ Series[ Sum[ MoebiusMu[k] k x^k/(1 - x^k), {k, 1, nmax} ], {x, 0, nmax} ], x ], 1 ] (* Stuart Clary, Apr 15 2006 *)
t[n_, 1] = 1; t[1, k_] = 1; t[n_, k_] :=  t[n, k] = If[n < k, If[n > 1 && k > 1, Sum[-t[k - i, n], {i, 1, n - 1}], 0], If[n > 1 && k > 1, Sum[-t[n - i, k], {i, 1, k - 1}], 0]]; Table[t[n, n], {n, 36}] (* Mats Granvik, Robert G. Wilson v, Jun 25 2011 *)
Table[DivisorSum[m, # MoebiusMu[#] &], {m, 90}] (* Jan Mangaldan, Mar 15 2013 *)
f[p_, e_] := (1 - p); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 14 2020 *)

{a(n) = direuler( p=2, n, (1 - p*X) / (1 - X))[n]}

{a(n) = if( n<1, 0, sumdiv( n, d, d * moebius(d)))} /* Michael Somos, Jul 18 2011 */

a(n)=sumdivmult(n,d, d*moebius(d)) \\ Charles R Greathouse IV, Sep 09 2014

from sympy import divisors, mobius
def a(n): return sum([d*mobius(d) for d in divisors(n)]) # Indranil Ghosh, Apr 29 2017

from math import prod
from sympy import primefactors
def A023900(n): return prod(1-p for p in primefactors(n)) # Chai Wah Wu, Sep 08 2023

;; With memoization-macro definec.
(definec (A023900 n) (if (= 1 n) 1 (* (- 1 (A020639 n)) (A023900 (A028234 n))))) ;; Antti Karttunen, Nov 28 2017

a(n) = Sum_{ d divides n } d*mu(d) = Product_{p|n} (1-p).
a(n) = 1 / (Sum_{ d divides n } mu(d)*d/phi(d)).
Dirichlet g.f.: zeta(s)/zeta(s-1). - Michael Somos, Jun 04 2000
a(n+1) = det(n+1)/det(n) where det(n) is the determinant of the n X n matrix M_(i, j) = i/gcd(i, j) = lcm(i, j)/j. - Benoit Cloitre, Aug 19 2003
a(n) = phi(n)*moebius(A007947(n))*A007947(n)/n. Logarithmic g.f.: Sum_{n >= 1} a(n)*x^n/n = log(F(x)) where F(x) is the g.f. of A117209 and satisfies: 1/(1-x) = Product_{n >= 1} F(x^n). - Paul D. Hanna, Mar 03 2006
G.f.: A(x) = Sum_{k >= 1} mu(k) k x^k/(1 - x^k) where mu(k) is the Moebius (Mobius) function, A008683. - Stuart Clary, Apr 15 2006
G.f.: A(x) is x times the logarithmic derivative of A117209(x). - Stuart Clary, Apr 15 2006
Row sums of triangle A134842. - Gary W. Adamson, Nov 12 2007
G.f.: x/(1-x) = Sum_{n >= 1} a(n)*x^n/(1-x^n)^2. - Paul D. Hanna, Aug 16 2008
a(n) = phi(rad(n)) *(-1)^omega(n) = A000010(A007947(n)) *(-1)^A001221(n). - Enrique Pérez Herrero, Aug 24 2010
a(n) = Product_{i = 2..n} (1-i)^( (pi(i)-pi(i-1)) * floor( (cos(n*Pi/i))^2 ) ), where pi = A000720, Pi = A000796. - Wesley Ivan Hurt, May 24 2013
a(n) = -limit of zeta(s)*(Sum_{d divides n} moebius(d)/exp(d)^(s-1)) as s->1 for n>1. - Mats Granvik, Jul 31 2013
a(n) = Sum_{d divides n} mu(d)*rad(d), where rad is A007947. - Enrique Pérez Herrero, May 29 2014
Conjecture for n>1: Let n = 2^(A007814(n))*m = 2^(ruler(n))*odd_part(n), where m = A000265(n), then a(n) = (-1)^(m=n)*(0+Sum_{i=1..m and gcd(i,m)=1} (4*min(i,m-i)-m)) = (-1)^(m1} (4*min(i,m-i)-m)). - I. V. Serov, May 02 2017
a(n) = (-1)^A001221(n) * A173557(n). - R. J. Mathar, Nov 02 2017
a(1) = 1; for n > 1, a(n) = (1-A020639(n)) * a(A028234(n)), because multiplicative with a(p^e) = (1-p). - Antti Karttunen, Nov 28 2017
a(n) = 1 - Sum_{d|n, d > 1} d*a(n/d). - Ilya Gutkovskiy, Apr 26 2019
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k)) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)) = 0. (End)
a(n) = rad(n)*(-1)^omega(n)*phi(n)/n = A062953(n)*A000010(n)/n. - Amrit Awasthi, Jan 30 2022
a(n) = mu(n)*phi(n) = A008683(n)*A000010(n) whenever n is squarefree. - Amrit Awasthi, Feb 03 2022
From Peter Bala, Jan 24 2024: (Start)
a(n) = Sum_{d divides n} core(d)*mu(d). Cf. Comment by Benoit Cloitre, dated Apr 06 2002.
a(n) = Sum_{d|n, e|n}  n/gcd(d, e) * mu(n/d) * mu(n/e) (the sum is a multiplicative function of n by Tóth, and takes the value 1 - p for n = p^e, a prime power). (End)
From Peter Bala, Feb 01 2024: (Start)
G.f. Sum_{n >= 1} (2*n-1)*moebius(2*n-1)*x^(2*n-1)/(1 + x^(2n-1)).
a(n) = (-1)^(n+1) * Sum_{d divides n, d odd} d*moebius(d). (End)

A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

Original entry on oeis.org

1, 3, 5, 8, 9, 15, 13, 20, 21, 27, 21, 40, 25, 39, 45, 48, 33, 63, 37, 72, 65, 63, 45, 100, 65, 75, 81, 104, 57, 135, 61, 112, 105, 99, 117, 168, 73, 111, 125, 180, 81, 195, 85, 168, 189, 135, 93, 240, 133, 195, 165, 200, 105, 243, 189, 260, 185, 171, 117, 360

Offset: 1

David W. Wilson

a(n) is the number of times the number 1 appears in the character table of the cyclic group C_n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 02 2001
a(n) is the number of ways to express all fractions f/g whereby each product (f/g)*n is a natural number between 1 and n (using fractions of the form f/g with 1 <= f,g <= n). For example, for n=4 there are 8 such fractions: 1/1, 1/2, 2/2, 3/3, 1/4, 2/4, 3/4 and 4/4. - Ron Lalonde (ronronronlalonde(AT)hotmail.com), Oct 03 2002
Number of non-congruent solutions to xy == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003
Conjecture: n>1 divides a(n)+1 iff n is prime. - Thomas Ordowski, Oct 22 2014
The above conjecture is false, with counterexample given by n = 3*37*43*42307*116341 and a(n)+1 = 26*n. - Varun Vejalla, Jun 19 2025
a(n) is the number of 0's in the multiplication table Z/nZ (cf. A000010 for number of 1's). - Eric Desbiaux, Jun 11 2015
{a(n)} == 1, 3, 1, 0, 1, 3, 1, 0, ... (mod 4). - Isaac Saffold, Dec 30 2017
Since a(p^e) = p^(e-1)*((p-1)e+p) it follows a(p) = 2p-1 and therefore p divides a(p)+1. - Ruediger Jehn, Jun 23 2022

			G.f. = x + 3*x^2 + 5*x^3 + 8*x^4 + 9*x^5 + 15*x^6 + 13*x^7 + 20*x^8 + ...

S. S. Pillai, On an arithmetic function, J. Annamalai University 2 (1933), pp. 243-248.
J. Sándor, A generalized Pillai function, Octogon Mathematical Magazine Vol. 9, No. 2 (2001), 746-748.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)
U. Abel, W. Awan, and V. Kushnirevych, A Generalization of the Gcd-Sum Function, J. Int. Seq. 16 (2013) #13.6.7.
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Olivier Bordelles, The Composition of the gcd and Certain Arithmetic Functions, J. Int. Seq. 13 (2010) #10.7.1.
Olivier Bordelles, An Asymptotic Formula for Short Sums of Gcd-Sum Functions, Journal of Integer Sequences, Vol. 15 (2012), #12.6.8.
Olivier Bordelles, A Multidimensional Cesáro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.
Kevin A. Broughan, The gcd-sum function, Journal of Integer Sequences 4 (2001), Article 01.2.2, 19 pp.
J.-M. De Koninck and I. Kátai, Some remarks on a paper of L. Toth, JIS 13 (2010) 10.1.2.
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Soichi Ikeda and Kaneaki Matsuoka, On the Lcm-Sum Function, Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.7.
Mathematical Reflections, Solution to Problem O364, Issue 2, 2016, p 24. [Cached copy at Wayback Machine]
Taylor McAdam, Almost-primes in horospherical flows on the space of lattices, arXiv:1802.08764 [math.DS], 2018.
Taylor McAdam, Almost-prime times in horospherical flows on the space of lattices, Journal of Modern Dynamics (2019) Vol. 15, 277-327.
J. Ransford, A Sum of Gcd’s over Friable Numbers, JIS vol 19 (2016) # 16.3.2
Jeffrey Shallit, Problem E 2821, American Mathematical Monthly 87 (1980), 220.
Jeffrey Shallit, Solution, American Mathematical Monthly, 88 (1981), 444-445.
László Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.
László Tóth, A survey of gcd-sum functions, J. Integer Sequences 13 (2010), Article 10.8.1, 23 pp.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7.
D. Zhang and W. Zhai, Mean Values of a Class of Arithmetical Functions, J. Int. Seq. 14 (2011) #11.6.5.
D. Zhang and W. Zhai, On an Open Problem of Tóth, J. Int. Seq. 16 (2013) #13.6.5.

Column 1 of A343510 and A343516.
Cf. A080997, A080998 for rankings of the positive integers in terms of centrality, defined to be the average fraction of an integer that it shares with the other integers as a gcd, or A018804(n)/n^2, also A080999, a permutation of this sequence (A080999(n) = A018804(A080997(n))).
Cf. A185210, A010766, A054523, A127468, A050873, A078430, A095026, A227507, A000010, A344372, A272718 (partial sums), A368736 - A368741.

a018804 n = sum $ map (gcd n) [1..n]  -- Reinhard Zumkeller, Jul 16 2012

[&+[Gcd(n,k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Nov 14 2019

a:=n->sum(igcd(n,j),j=1..n): seq(a(n), n=1..60); # Zerinvary Lajos, Nov 05 2006

f[n_] := Block[{d = Divisors[n]}, Sum[ d*EulerPhi[n/d], {d, d}]]; Table[f[n], {n, 60}] (* Robert G. Wilson v, Mar 20 2012 *)
a[ n_] := If[ n < 1, 0, n Sum[ EulerPhi[d] / d, {d, Divisors@n}]]; (* Michael Somos, Jan 07 2017 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 19 2019 *)

{a(n) = direuler(p=2, n, (1 - X) / (1 - p*X)^2)[n]}; /* Michael Somos, May 31 2000 */

a(n)={ my(ct=0); for(i=0,n-1,for(j=0,n-1, ct+=(Mod(i*j,n)==0) ) ); ct; } \\ Joerg Arndt, Aug 03 2013

a(n)=my(f=factor(n)); prod(i=1,#f~,(f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]) \\ Charles R Greathouse IV, Oct 28 2014

a(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ Michel Marcus, Jan 07 2017

from sympy.ntheory import totient, divisors
print([sum(n*totient(d)//d for d in divisors(n)) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

from sympy import factorint
from math import prod
def A018804(n): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, Nov 29 2021

a(n) = Sum_{d|n} d*phi(n/d), where phi(n) is Euler totient function (cf. A000010). - Vladeta Jovovic, Apr 04 2001
Multiplicative; for prime p, a(p^e) = p^(e-1)*((p-1)e+p).
Dirichlet g.f.: zeta(s-1)^2/zeta(s).
a(n) = Sum_{d|n} d*tau(d)*mu(n/d). - Benoit Cloitre, Oct 23 2003
Equals A054523 * [1,2,3,...]. Equals row sums of triangle A010766. - Gary W. Adamson, May 20 2007
Equals inverse Mobius transform of A029935 = A054525 * (1, 2, 4, 5, 8, 8, 12, 12, ...). - Gary W. Adamson, Aug 02 2008, corrected Feb 07 2023
Equals row sums of triangle A127478. - Gary W. Adamson, Aug 03 2008
G.f.: Sum_{k>=1} phi(k)*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{a = 1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1, for n > 1. The sum is over a,b,c such that n*c - a*b = 0. - Benedict W. J. Irwin, Apr 04 2017
Proof: Let gcd(a, n) = g and x = n/g. Define B = {x, 2*x, ..., g*x}; then for all b in B there exists a number c such that a*b = n*c. Since the set B has g elements it follows that Sum_{b=1..n} Sum_{c=1..n} 1 >= g = gcd(a, n) and therefore Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 >= Sum_{a=1..n} gcd(a, n). On the other hand, for all b not in B there is no number c <= n such that a*b = n*c and hence Sum_{b = 1..n} Sum_{c = 1..n} 1 = g. Therefore Sum_{a=1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1 = a(n). - Ruediger Jehn, Jun 23 2022
a(2*n) = a(n)*(3-A007814(n)/(A007814(n)+2)) - Velin Yanev, Jun 30 2017
Proof: Let m = A007814(m) and decompose n into n = k*2^m. We know from Chai Wah Wu's program below that a(n) = Product(p_i^(e_i-1)*((p_i-1)*e_i+p_i)) where the numbers p_i are the prime factors of n and e_i are the corresponding exponents. Hence a(2n) = 2^m*(m+3)*a(k) = 2^m*(m+3)*a(k). On the other hand, a(n) = 2^(m-1)*(m+2)*a(k). Dividing the first equation by the second yields a(2n)/a(n) = 2*(m+3)/(m+2), which equals 3 - m/(m+2). Hence a(2n) = a(n)*(3 - m/(m+2)). - Ruediger Jehn, Jun 23 2022
Sum_{k=1..n} a(k) ~ 3*n^2/Pi^2 * (log(n) - 1/2 + 2*gamma - 6*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 08 2019
a(n) = Sum_{k=1..n} n/gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021
log(a(n)/n) << log n log log log n/log log n; in particular, a(n) << n^(1+e) for any e > 0. See Broughan link for bounds in terms of omega(n). - Charles R Greathouse IV, Sep 08 2022
a(n) = (1/4)*Sum_{k = 1..4*n} (-1)^k * gcd(k, 4*n) = (1/4) * A344372(2*n). - Peter Bala, Jan 01 2024

A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 210, 8, 36, 60, 2310, 24, 30030, 420, 180, 16, 510510, 72, 9699690, 120, 1260, 4620, 223092870, 48, 900, 60060, 216, 840, 6469693230, 360, 200560490130, 32, 13860, 1021020, 6300, 144, 7420738134810, 19399380, 180180, 240, 304250263527210, 2520

Offset: 1

Paul Boddington, Jul 21 2005

This sequence is a permutation of A025487.
And thus also a permutation of A181812, see the formula section. - Antti Karttunen, Jul 21 2014
A previous description of this sequence was: "Multiplicative with a(p^e) equal to the product of the e-th powers of all primes at most p" (see extensions), Giuseppe Coppoletta, Feb 28 2015

			a(12) = a(2^2) * a(3) = (2#)^2 * (3#) = 2^2 * 6 = 24
a(45) = (3#)^2 * (5#) = (2*3)^2 * (2*3*5) = 1080 (as 45 = 3^2 * 5).

Amiram Eldar, Table of n, a(n) for n = 1..2370 (terms 1..256 from Antti Karttunen)
Index to divisibility sequences.
Index entries for sequences computed from indices in prime factorization.
Index entries for sequences related to primorial numbers.

Cf. A319626, A329900 (left inverses).

Cf. A034386, A002110, A025487, A048673, A064216, A064989, A085082, A122111, A124859, A161360, A181811, A181812, A181814, A181815, A181817, A181819, A181822, A238690, A283477, A283478, A307035, A324886, A324887, A324888, A324896, A325226, A329040, A329046, A329047, A329344, A329348, A329349, A329378, A329382, A329600, A329602, A329605, A329607, A329615, A329616, A329617, A329619, A329622, A319627, A329647, A331292, A337474, A346108, A346109, A344698, A344699.

a[n_] := a[n] = Module[{f = FactorInteger[n], p, e}, If[Length[f]>1, Times @@ a /@ Power @@@ f, {{p, e}} = f; Times @@ (Prime[Range[PrimePi[p]]]^e)]]; a[1] = 1; Table[a[n], {n, 1, 42}] (* Jean-François Alcover, Feb 24 2015 *)
Table[Times @@ Map[#1^#2 & @@ # &, FactorInteger[n] /. {p_, e_} /; e > 0 :> {Times @@ Prime@ Range@ PrimePi@ p, e}], {n, 42}] (* Michael De Vlieger, Mar 18 2017 *)

primorial(n)=prod(i=1,primepi(n),prime(i))
a(n)=my(f=factor(n)); prod(i=1,#f~, primorial(f[i,1])^f[i,2]) \\ Charles R Greathouse IV, Jun 28 2015

from sympy import primerange, factorint
from operator import mul
def P(n): return reduce(mul, [i for i in primerange(2, n + 1)])
def a(n):
    f = factorint(n)
    return 1 if n==1 else reduce(mul, [P(i)**f[i] for i in f])
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, May 14 2017

def sharp_primorial(n): return sloane.A002110(prime_pi(n))
def p(f):
    return sharp_primorial(f[0])^f[1]
[prod(p(f) for f in factor(n)) for n in range (1,51)]
# Giuseppe Coppoletta, Feb 07 2015

Dirichlet g.f.: 1/(1-2*2^(-s))/(1-6*3^(-s))/(1-30*5^(-s))...
Completely multiplicative with a(p_i) = A002110(i) = prime(i)#. [Franklin T. Adams-Watters, Jun 24 2009; typos corrected by Antti Karttunen, Jul 21 2014]
From Antti Karttunen, Jul 21 2014: (Start)
a(1) = 1, and for n > 1, a(n) = n * a(A064989(n)).
a(n) = n * A181811(n).
a(n) = A002110(A061395(n)) * A331188(n). - [added Jan 14 2020]
a(n) = A181812(A048673(n)).
Other identities:
A006530(a(n)) = A006530(n). [Preserves the largest prime factor of n.]
A071178(a(n)) = A071178(n). [And also its exponent.]
a(2^n) = 2^n. [Fixes the powers of two.]
A067029(a(n)) = A007814(a(n)) = A001222(n). [The exponent of the least prime of a(n), that prime always being 2 for n>1, is equal to the total number of prime factors in n.]
(End)
From Antti Karttunen, Nov 19 2019: (Start)
Further identities:
a(A307035(n)) = A000142(n).
a(A003418(n)) = A181814(n).
a(A025487(n)) = A181817(n).
a(A181820(n)) = A181822(n).
a(A019565(n)) = A283477(n).
A001221(a(n)) = A061395(n).
A001222(a(n)) = A056239(n).
A181819(a(n)) = A122111(n).
A124859(a(n)) = A181821(n).
A085082(a(n)) = A238690(n).
A328400(a(n)) = A329600(n). (smallest number with the same set of distinct prime exponents)
A000188(a(n)) = A329602(n). (square root of the greatest square divisor)
A072411(a(n)) = A329378(n). (LCM of exponents of prime factors)
A005361(a(n)) = A329382(n). (product of exponents of prime factors)
A290107(a(n)) = A329617(n). (product of distinct exponents of prime factors)
A000005(a(n)) = A329605(n). (number of divisors)
A071187(a(n)) = A329614(n). (smallest prime factor of number of divisors)
A267115(a(n)) = A329615(n). (bitwise-AND of exponents of prime factors)
A267116(a(n)) = A329616(n). (bitwise-OR of exponents of prime factors)
A268387(a(n)) = A329647(n). (bitwise-XOR of exponents of prime factors)
A276086(a(n)) = A324886(n). (prime product form of primorial base expansion)
A324580(a(n)) = A324887(n).
A276150(a(n)) = A324888(n). (digit sum in primorial base)
A267263(a(n)) = A329040(n). (number of distinct nonzero digits in primorial base)
A243055(a(n)) = A329343(n).
A276088(a(n)) = A329348(n). (least significant nonzero digit in primorial base)
A276153(a(n)) = A329349(n). (most significant nonzero digit in primorial base)
A328114(a(n)) = A329344(n). (maximal digit in primorial base)
A062977(a(n)) = A325226(n).
A097248(a(n)) = A283478(n).
A324895(a(n)) = A324896(n).
A324655(a(n)) = A329046(n).
A327860(a(n)) = A329047(n).
A329601(a(n)) = A329607(n).
(End)
a(A181815(n)) = A025487(n), and A319626(a(n)) = A329900(a(n)) = n. - Antti Karttunen, Dec 29 2019
From Antti Karttunen, Jul 09 2021: (Start)
a(n) = A346092(n) + A346093(n).
a(n) = A346108(n) - A346109(n).
a(A342012(n)) = A004490(n).
a(A337478(n)) = A336389(n).
A336835(a(n)) = A337474(n).
A342002(a(n)) = A342920(n).
A328571(a(n)) = A346091(n).
A328572(a(n)) = A344592(n).
(End)
Sum_{n>=1} 1/a(n) = A161360. - Amiram Eldar, Aug 04 2022

More terms computed by Antti Karttunen, Jul 21 2014
The name of the sequence was changed for more clarity, in accordance with the above remark of Franklin T. Adams-Watters (dated Jun 24 2009). It is implicitly understood that a(n) is then uniquely defined by completely multiplicative extension. - Giuseppe Coppoletta, Feb 28 2015
Name "Primorial inflation" (coined by Matthew Vandermast in A181815) prefixed to the name by Antti Karttunen, Jan 14 2020

A048250 Sum of the squarefree divisors of n.

Original entry on oeis.org

1, 3, 4, 3, 6, 12, 8, 3, 4, 18, 12, 12, 14, 24, 24, 3, 18, 12, 20, 18, 32, 36, 24, 12, 6, 42, 4, 24, 30, 72, 32, 3, 48, 54, 48, 12, 38, 60, 56, 18, 42, 96, 44, 36, 24, 72, 48, 12, 8, 18, 72, 42, 54, 12, 72, 24, 80, 90, 60, 72, 62, 96, 32, 3, 84, 144, 68, 54, 96, 144, 72, 12, 74

Offset: 1

Labos Elemer

Also sum of divisors of the squarefree kernel of n: a(n) = A000203(A007947(n)). - Reinhard Zumkeller, Jul 19 2002
The absolute values of the Dirichlet inverse of A001615. - R. J. Mathar, Dec 22 2010
Row sums of the triangle in A206778. - Reinhard Zumkeller, Feb 12 2012
Inverse Möbius transform of n * mu(n)^2 = |A055615(n)|. - Wesley Ivan Hurt, Jun 08 2023

			For n=1000, out of the 16 divisors, four are squarefree: {1,2,5,10}. Their sum is 18. Or, 1000 = 2^3*5^3 hence a(1000) = (2+1)*(5+1) = 18.

D. Suryanarayana, On the core of an integer, Indian J. Math. 14 (1972) 65-74.

T. D. Noe, Table of n, a(n) for n = 1..10000
Steven R. Finch, Unitarism and infinitarism.
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]
Index entries for sequences related to sums of squares

Cf. A003557, A007913, A007947, A023900, A034448, A206787, A309192.
Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), this sequence (k=1), A351265 (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
Cf. A240976 (tenth of Dgf at s=3).

a034448 = sum . a206778_row  -- Reinhard Zumkeller, Feb 12 2012

A048250 := proc(n) local ans, i:ans := 1: for i from 1 to nops(ifactors(n)[ 2 ]) do ans := ans*(1+ifactors(n)[ 2 ][ i ] [ 1 ]): od: RETURN(ans) end:
# alternative:
seq(mul(1+p, p = numtheory:-factorset(n)), n=1..1000); # Robert Israel, Mar 18 2015

sumOfSquareFreeDivisors[ n_ ] := Plus @@ Select[ Divisors[ n ], MoebiusMu[ # ] != 0 & ]; Table[ sumOfSquareFreeDivisors[ i ], {i, 85} ]
Table[Total[Select[Divisors[n],SquareFreeQ]],{n,80}] (* Harvey P. Dale, Jan 25 2013 *)
a[1] = 1; a[n_] := Times@@(1 + FactorInteger[n][[;;,1]]); Array[a, 100] (* Amiram Eldar, Dec 19 2018 *)

a(n)=if(n<1,0,sumdiv(n,d,if(core(d)==d,d)))

a(n)=if(n<1,0,direuler(p=2,n,(1+p*X)/(1-X))[n])

a(n)=sumdiv(n,d,moebius(d)^2*d); \\ Joerg Arndt, Jul 06 2011

a(n)=my(f=factor(n)); for(i=1,#f~,f[i,2]=1); sigma(f) \\ Charles R Greathouse IV, Sep 09 2014

from math import prod
from sympy import primefactors
def A048250(n): return prod(p+1 for p in primefactors(n)) # Chai Wah Wu, Apr 20 2023

def A048250(n): return mul(map(lambda p: p+1, prime_divisors(n)))
[A048250(n) for n in (1..73)]  # Peter Luschny, May 23 2013

If n = Product p_i^e_i, a(n) = Product (p_i + 1). - Vladeta Jovovic, Apr 19 2001
Dirichlet g.f.: zeta(s)*zeta(s-1)/zeta(2*s-2). - Michael Somos, Sep 08 2002
a(n) = Sum_{d|n} mu(d)^2*d = Sum_{d|n} |A055615(d)|. - Benoit Cloitre, Dec 09 2002
Pieter Moree (moree(AT)mpim-bonn.mpg.de), Feb 20 2004 can show that Sum_{n <= x} a(n) = x^2/2 + O(x*sqrt{x}) and adds: "As S. R. Finch pointed out to me, in Suryanarayana's paper this is proved under the Riemann hypothesis with error term O(x^{7/5+epsilon})".
a(n) = psi(rad(n)) = A001615(A007947(n)). - Enrique Pérez Herrero, Aug 24 2010
a(n) = rad(n)*psi(n)/n = A001615(n)*A007947(n)/n. - Enrique Pérez Herrero, Aug 31 2010
G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - Ilya Gutkovskiy, Jan 03 2017
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = 1. - Amiram Eldar, Jun 10 2020
a(n) = Sum_{d divides n} mu(d)^2*core(d), where core(n) = A007913(n). - Peter Bala, Jan 24 2024

A000188 (1) Number of solutions to x^2 == 0 (mod n). (2) Also square root of largest square dividing n. (3) Also max_{ d divides n } gcd(d, n/d).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 4, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 4, 1, 1, 1, 6, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 4, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 8, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 5, 2, 1, 1, 1, 4, 9, 1, 1, 2, 1, 1, 1, 2, 1, 3

Offset: 1

N. J. A. Sloane

Shadow transform of the squares A000290. - Vladeta Jovovic, Aug 02 2002
Labos Elemer and Henry Bottomley independently proved that (2) and (3) define the same sequence. Bottomley also showed that (1) and (2) define the same sequence.
Proof that (2) = (3): Let max{gcd(d, n/d)} = K, then d = Kx, n/d = Ky so n = KKxy where xy is the squarefree part of n, otherwise K is not maximal. Observe also that g = gcd(K, xy) is not necessarily 1. Thus K is also the "maximal square-root factor" of n. - Labos Elemer, Jul 2000
We can write sqrt(n) = b*sqrt(c) where c is squarefree. Then b = A000188(n) is the "inner square root" of n, c = A007913(n) and b*c = A019554(n) = "outer square root" of n.

			a(8) = 2 because the largest square dividing 8 is 4, the square root of which is 2.
a(9) = 3 because 9 is a perfect square and its square root is 3.
a(10) = 1 because 10 is squarefree.

T. D. Noe, Table of n, a(n) for n = 1..10000
Henry Bottomley, Some Smarandache-type multiplicative sequences.
Kevin A. Broughan, Restricted divisor sums, preprint.
Kevin A. Broughan, Restricted divisor sums, Acta Arithmetica, 101(2) (2002), 105-114.
Kevin A. Broughan, Relationship between the integer conductor and k-th root functions, Int. J. Pure Appl. Math. 5(3) (2003), 253-275.
Kevin A. Broughan, Relaxations of the ABC Conjecture using integer k'th roots, New Zealand J. Math. 35(2) (2006), 121-136.
John M. Campbell, An Integral Representation of Kekulé Numbers, and Double Integrals Related to Smarandache Sequences, arXiv preprint arXiv:1105.3399 [math.GM], 2011.
Steven R. Finch and Pascal Sebah, Squares and Cubes Modulo n, arXiv:math/0604465 [math.NT], 2006-2016.
Vaclav Kotesovec, Graph - the asymptotic ratio (100000 terms)
Gerry Myerson, Trifectas in Geometric Progression, Australian Mathematical Society Gazette 35(3) (2008), 189-194
Andrew Reiter, On (mod n) spirals (2014), and posting to Number Theory Mailing List, Mar 23 2014.
N. J. A. Sloane, Transforms.
Florentin Smarandache, Collected Papers, Vol. II, Tempus Publ. Hse, Bucharest, 1996.
László Tóth, Counting solutions of quadratic congruences in several variables revisited, arXiv preprint arXiv:1404.4214 [math.NT], 2014.
László Tóth, Counting Solutions of Quadratic Congruences in Several Variables Revisited, J. Int. Seq. 17 (2014), # 14.11.6.

Cf. A019554 (outer square root), A053150 (inner 3rd root), A019555 (outer 3rd root), A053164 (inner 4th root), A053166 (outer 4th root), A015052 (outer 5th root), A015053 (outer 6th root).
Cf. A000189, A000190, A007947, A008833, A027748, A046951, A055210, A007913, A117811, A124010. For partial sums see A120486.
Cf. A240976 (Dgf at s=2).

a000188 n = product $ zipWith (^)
                      (a027748_row n) $ map (`div` 2) (a124010_row n)
-- Reinhard Zumkeller, Apr 22 2012

with(numtheory):A000188 := proc(n) local i: RETURN(op(mul(i,i=map(x->x[1]^floor(x[2]/2),ifactors(n)[2])))); end;

Array[Function[n, Count[Array[PowerMod[#, 2, n ] &, n, 0 ], 0 ] ], 100]
(* Second program: *)
nMax = 90; sList = Range[Floor[Sqrt[nMax]]]^2; Sqrt[#] &/@ Table[ Last[ Select[ sList, Divisible[n, #] &]], {n, nMax}] (* Harvey P. Dale, May 11 2011 *)
a[n_] := With[{d = Divisors[n]}, Max[GCD[d, Reverse[d]]]] (* Mamuka Jibladze, Feb 15 2015 *)
f[p_, e_] := p^Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

PARI
```
a(n)=if(n<1,0,sum(i=1,n,i*i%n==0))
```

a(n)=sqrtint(n/core(n)) \\ Zak Seidov, Apr 07 2009

a(n)=core(n, 1)[2] \\ Michel Marcus, Feb 27 2013

from sympy.ntheory.factor_ import core
from sympy import integer_nthroot
def A000188(n): return integer_nthroot(n//core(n),2)[0] # Chai Wah Wu, Jun 14 2021

a(n) = n/A019554(n) = sqrt(A008833(n)).
a(n) = Sum_{d^2|n} phi(d), where phi is the Euler totient function A000010.
Multiplicative with a(p^e) = p^floor(e/2). - David W. Wilson, Aug 01 2001
Dirichlet series: Sum_{n >= 1} a(n)/n^s = zeta(2*s - 1)*zeta(s)/zeta(2*s), (Re(s) > 1).
Dirichlet convolution of A037213 and A008966. - R. J. Mathar, Feb 27 2011
Finch & Sebah show that the average order of a(n) is 3 log n/Pi^2. - Charles R Greathouse IV, Jan 03 2013
a(n) = sqrt(n/A007913(n)). - M. F. Hasler, May 08 2014
Sum_{n>=1} lambda(n)*a(n)*x^n/(1-x^n) = Sum_{n>=1} n*x^(n^2), where lambda() is the Liouville function A008836 (cf. A205801). - Mamuka Jibladze, Feb 15 2015
a(2*n) = a(n)*(A096268(n-1) + 1). - observed by Velin Yanev, Jul 14 2017, The formula says that a(2n) = 2*a(n) only when 2-adic valuation of n (A007814(n)) is odd, otherwise a(2n) = a(n). This follows easily from the definition (2). - Antti Karttunen, Nov 28 2017
Sum_{k=1..n} a(k) ~ 3*n*((log(n) + 3*gamma - 1)/Pi^2 - 12*zeta'(2)/Pi^4), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Dec 01 2020
Conjecture: a(n) = Sum_{k=1..n} A010052(n*k). - Velin Yanev, Jul 04 2021
G.f.: Sum_{k>=1} phi(k) * x^(k^2) / (1 - x^(k^2)). - Ilya Gutkovskiy, Aug 20 2021

Edited by M. F. Hasler, May 08 2014

A063524 Characteristic function of 1.

Original entry on oeis.org

0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Labos Elemer, Jul 30 2001

The identity function for Dirichlet multiplication (see Apostol).
Sum of the Moebius function mu(d) of the divisors d of n. - Robert G. Wilson v, Sep 30 2006
-a(n) is the Hankel transform of A000045(n), n >= 0 (Fibonacci numbers). See A055879 for the definition of Hankel transform. - Wolfdieter Lang, Jan 23 2007
a(A000012(n)) = 1; a(A087156(n)) = 0. - Reinhard Zumkeller, Oct 11 2008
a(n) for n >= 1 is the Dirichlet convolution of following functions b(n), c(n), a(n) = Sum_{d|n} b(d)*c(n/d): a(n) = A008683(n) * A000012(n), a(n) = A007427(n) * A000005(n), a(n) = A007428(n) * A007425(n). - Jaroslav Krizek, Mar 03 2009
From Christopher Hunt Gribble, Jul 11 2013: (Start)
a(n) for 1 <= n <= 4 and conjectured for n > 4 is the number of Hamiltonian circuits in a 2n X 2n square lattice of nodes, reduced for symmetry, where the orbits under the symmetry group of the square, D4, have 1 element: When n=1, there is only 1 Hamiltonian circuit in a 2 X 2 square lattice, as illustrated below.  The circuit is the same when rotated and/or reflected and so has only 1 orbital element under the symmetry group of the square.
     o--o
     |  |
     o--o  (End)
Convolution property: For any sequence b(n), the sequence c(n)=b(n)*a(n) has the following values: c(1)=0, c(n+1)=b(n) for all n > 1. In other words, the sequence b(n) is shifted 1 step to the right. - David Neil McGrath, Nov 10 2014

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 30.

Antti Karttunen, Table of n, a(n) for n = 0..100000
G. P. Michon, Multiplicative Functions
Wikipedia, Dirichlet convolution
Index entries for sequences computed from exponents in factorization of n
Index entries for linear recurrences with constant coefficients, signature (1).
Index entries for characteristic functions

Cf. A000007 (same sequence shifted once left).

Cf. A000005, A000010, A000012, A000027, A003763, A008683, A007427, A007428, A007425, A023900, A055615, A101296, A227005, A227257, A227301, A209077.

a063524 = fromEnum . (== 1)  -- Reinhard Zumkeller, Apr 01 2012

A063524 := proc(n) if n = 1 then 1 else 0; fi; end;

Table[If[n == 1, 1, 0], {n, 0, 104}] (* Robert G. Wilson v, Sep 30 2006 *)
LinearRecurrence[{1},{0,1,0},106] (* Ray Chandler, Jul 15 2015 *)

a(n)=n==1; \\ Charles R Greathouse IV, Apr 01 2012

def A063524(n): return int(n==1) # Chai Wah Wu, Feb 04 2022

From Philippe Deléham, Nov 25 2008: (Start)
G.f.: x.
E.g.f.: x. (End)
a(n) = mu(n^2). - Enrique Pérez Herrero, Sep 04 2009
a(n) = floor(n/A000203(n)) for n > 0. - Enrique Pérez Herrero, Nov 11 2009
a(n) = (1-(-1)^(2^abs(n-1)))/2 = (1-(-1)^(2^((n-1)^2)))/2. - Luce ETIENNE, Jun 05 2015
a(n) = n*(A057427(n) - A057427(n-1)) = A000007(abs(n-1)). - Chayim Lowen, Aug 01 2015
a(n) = A010051(p*n) for any prime p (where A010051(0)=0). - Chayim Lowen, Aug 05 2015
From Antti Karttunen, Jun 04 2022: (Start)
For n >= 1:
a(n) = Sum_{d|n} A000010(n/d) * A023900(d), and similarly for any pair of sequences that are Dirichlet inverses of each other, like for example A000027 & A055615 and those mentioned in Krizek's Mar 03 2009 comment above.
a(n) = [A101296(n) == 1], where [ ] is the Iverson bracket.
Fully multiplicative with a(p^e) = 0. (End)

A001160 sigma_5(n), the sum of the 5th powers of the divisors of n.

Original entry on oeis.org

1, 33, 244, 1057, 3126, 8052, 16808, 33825, 59293, 103158, 161052, 257908, 371294, 554664, 762744, 1082401, 1419858, 1956669, 2476100, 3304182, 4101152, 5314716, 6436344, 8253300, 9768751, 12252702, 14408200, 17766056, 20511150

Offset: 1

N. J. A. Sloane

If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/504. - Simon Plouffe, Mar 01 2021

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 166.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Zagier, Don. "Elliptic modular forms and their applications." The 1-2-3 of modular forms. Springer Berlin Heidelberg, 2008. 1-103. See p. 17, G_6(z).

T. D. Noe, Table of n, a(n) for n = 1..10000
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 827.
Index entries for sequences related to sigma(n)

Cf. A000005, A000203, A001157, A001158, A001159, A013973, A000584 (Mobius transform), A178448 (Dirichlet inverse)

[DivisorSigma(5,n): n in [1..30]]; // Bruno Berselli, Apr 10 2013

A001160 := proc(n)
    numtheory[sigma][5](n);
end proc:
seq(A001160(n),n=1..30) ; # R. J. Mathar, Jan 31 2017

lst={};Do[AppendTo[lst,DivisorSigma[5,n]],{n,5!}];lst (* Vladimir Joseph Stephan Orlovsky, Mar 11 2009 *)
DivisorSigma[5,Range[30]] (* Harvey P. Dale, Nov 11 2013 *)

a(n)=sigma(n,5) \\ Charles R Greathouse IV, Apr 28 2011

[sigma(n, 5) for n in range(1, 30)]  # Zerinvary Lajos, Jun 04 2009

Multiplicative with a(p^e) = (p^(5e+5)-1)/(p^5-1). - David W. Wilson, Aug 01 2001
G.f.: sum(k>=1, k^5*x^k/(1-x^k)). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s)*zeta(s-5). - R. J. Mathar, Mar 06 2011
G.f. also (1 - E_6(q))/540, with the g.f. E_6 of A013973. See Hardy p. 166, (10.5.7) with R = E_6. - Wolfdieter Lang, Jan 31 2017
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^4)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
a(n) = Sum_{1 <= i, j, k, l, m <= n} tau(gcd(i, j, k, l, m, n)) = Sum_{d divides n} tau(d) * J_5(n/d), where the divisor function tau(n) = A000005(n) and the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 22 2024

A007425 d_3(n), or tau_3(n), the number of ordered factorizations of n as n = r s t.

Original entry on oeis.org

1, 3, 3, 6, 3, 9, 3, 10, 6, 9, 3, 18, 3, 9, 9, 15, 3, 18, 3, 18, 9, 9, 3, 30, 6, 9, 10, 18, 3, 27, 3, 21, 9, 9, 9, 36, 3, 9, 9, 30, 3, 27, 3, 18, 18, 9, 3, 45, 6, 18, 9, 18, 3, 30, 9, 30, 9, 9, 3, 54, 3, 9, 18, 28, 9, 27, 3, 18, 9, 27, 3, 60, 3, 9, 18, 18, 9, 27, 3, 45, 15, 9, 3, 54, 9, 9, 9, 30, 3

Offset: 1

N. J. A. Sloane, May 24 1994

Let n = Product p_i^e_i. Tau (A000005) is tau_2, this sequence is tau_3, A007426 is tau_4, where tau_k(n) (also written as d_k(n)) = Product_i binomial(k-1+e_i, k-1) is the k-th Piltz function. It gives the number of ordered factorizations of n as a product of k terms. - Len Smiley
Inverse Möbius transform applied twice to all 1's sequence.
A085782 gives the range of values of this sequence. - Matthew Vandermast, Jul 12 2004
Appears to equal the number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. - Wouter Meeussen, Sep 11 2004
Number of divisors of n's divisors. - Lekraj Beedassy, Sep 07 2004
Number of plane partitions of n that can be extended in exactly 3 ways to a plane partition of n+1 by adding one element. If the partition is not a box, there is a minimal i+j where b_{i,j} != b_{1,1} and an element can be added there. - Franklin T. Adams-Watters, Jun 14 2006
Equals row sums of A127170. - Gary W. Adamson, May 20 2007
Equals A134577 * [1/1, 1/2, 1/3, ...]. - Gary W. Adamson, Nov 02 2007
Equals row sums of triangle A143354. - Gary W. Adamson, Aug 10 2008
a(n) is congruent to 1 (mod 3) if n is a perfect cube, otherwise a(n) is congruent to 0 (mod 3). - Geoffrey Critzer, Mar 20 2015
Also row sums of A195050. - Omar E. Pol, Nov 26 2015
Number of 3D grids of n congruent boxes with three different edge lengths, in a box, modulo rotation (cf. A034836 for cubes instead of boxes and A140773 for boxes with two different edge lengths; cf. A000005 for the 2D case). - Manfred Boergens, Apr 06 2021
Number of ordered pairs of divisors of n, (d1,d2) with d1<=d2, such that d1|d2. - Wesley Ivan Hurt, Mar 22 2022

			a(6) = 9; the divisors of 6 are {1,2,3,6} and the numbers of divisors of these divisors are 1, 2, 2, and 4. Adding them, we get 9 as a result.
Also, since 6 is a squarefree number, the formula from Herrero can be used to obtain the result: a(6) = 3^omega(6) = 3^2 = 9. - _Wesley Ivan Hurt_, May 30 2014

M. N. Huxley, Area, Lattice Points and Exponential Sums, Oxford, 1996; p. 239.
A. Ivic, The Riemann Zeta-Function, Wiley, NY, 1985, see p. xv.
Paul J. McCarthy, Introduction to Arithmetical Functions, Springer, 1986.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
R. Eisinga, R. Breitling, and T. Heskes, The exact probability distribution of the rank product statistics for replicated experiments, FEBS Letters, 2013, 587: 677-682, doi:10.1016/j.febslet.2013.01.037
Karin Cvetko-Vah, Michael Kinyon, Jonathan Leech, and Tomaž Pisanski, Regular Antilattices, arXiv:1911.02858 [math.RA], 2019.
Daniele A. Gewurz and Francesca Merola, Sequences realized as Parker vectors of oligomorphic permutation groups, J. Integer Seqs., Vol. 6, 2003.
Adolf Piltz, Ueber das Gesetz, nach welchem die mittlere Darstellbarkeit der natürlichen Zahlen als Produkte einer gegebenen Anzahl Faktoren mit der Grösse der Zahlen wächst, Doctoral Dissertation, Friedrich-Wilhelms-Universität zu Berlin, 1881; the k-th Piltz function tau_k(n) is denoted by phi(n,k) and its recurrence and Dirichlet series appear on p. 6.
N. J. A. Sloane, Transforms.
Qingfeng Sun and Deyu Zhang, Sums of the triple divisor function over values of a ternary quadratic form, arXiv:1510.06170 [math.NT], 2015.
E. C. Titchmarsh, Some problems in the analytic theory of numbers, The Quarterly Journal of Mathematics 1 (1942): 129-152.
Wikipedia, Adolf Piltz.

Cf. A000005 (Mobius transform), A007426 (inverse Mobius transform), A061201 (partial sums), A127270, A143354, A027750, A007428 (Dirichlet inverse), A175596.
Column k=3 of A077592.
Additional cross-references mentioned in a comment: A034836, A038548, A140733.

a007425 = sum . map a000005 . a027750_row
-- Reinhard Zumkeller, Feb 16 2012

f:=proc(n) local t1,i,j,k; t1:=0; for i from 1 to n do for j from 1 to n do for k from 1 to n do if i*j*k = n then t1:=t1+1; fi; od: od: od: t1; end;
A007425 := proc(n) local e,j; e := ifactors(n)[2]: product(binomial(2+e[j][2],2), j=1..nops(e)); end; # Len Smiley

f[n_] := Plus @@ DivisorSigma[0, Divisors[n]]; Table[ f[n], {n, 90}] (* Robert G. Wilson v, Sep 13 2004 *)
SetAttributes[tau, Listable]; tau[1, n_] := 1; tau[k_, n_] := Plus @@ (tau[k-1, Divisors[n]]); Table[tau[3, n], {n, 100}] (* Enrique Pérez Herrero, Nov 08 2009 *)
Table[Sum[DivisorSigma[0, d], {d, Divisors[n]}], {n, 50}] (* Wesley Ivan Hurt, May 30 2014 *)
f[p_, e_] := (e+1)*(e+2)/2;  a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 27 2019 *)

for(n=1,100,print1(sumdiv(n,k,numdiv(k)),","))

a(n)=if(n<1,0,direuler(p=2,n,1/(1-X)^3)[n]) \\ Ralf Stephan

a(n)=sumdiv(n, x, sumdiv(x, y, 1 )) \\ Joerg Arndt, Oct 07 2012

a(n)=sumdivmult(n,k,numdiv(k)) \\ Charles R Greathouse IV, Aug 30 2013

for(n=1, 100, print1(numerator(direuler(p=2, n, 1/(1-X)^3)[n]), ", ")) \\ Vaclav Kotesovec, May 06 2025

from math import prod, comb
from sympy import factorint
def A007425(n): return prod(comb(2+e,2) for e in factorint(n).values()) # Chai Wah Wu, Dec 22 2024

a(n) = Sum_{d dividing n} tau(d). - Benoit Cloitre, Apr 04 2002
G.f.: Sum_{k>=1} tau(k)*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
For n = Product p_i^e_i, a(n) = Product_i A000217(e_i + 1). - Lekraj Beedassy, Sep 07 2004
Dirichlet g.f.: zeta^3(s).
From Enrique Pérez Herrero, Nov 03 2009: (Start)
a(n^2) = tau_3(n^2) = tau_2(n^2)*tau_2(n), where tau_2 is A000005 and tau_3 is this sequence.
a(s) = 3^omega(s), if s>1 is squarefree (A005117) and omega(s) is: A001221. (End)
From Enrique Pérez Herrero, Nov 08 2009: (Start)
a(n) = tau_3(n) = tau_2(n)*tau_2(n*rad(n))/tau_2(rad(n)), where rad(n) is A007947 and tau_2(n) is A000005.
tau_3(n) >= 2*tau_2(n) - 1.
tau_3(n) <= tau_2(n)^2 + tau_2(n)-1. (End)
From Vladimir Shevelev, Dec 22 2017: (Start)
a(n) = sqrt(Sum_{d|n}(tau(d))^3);
a(n) = |Sum_{d|n} A008836(d)*(tau(d))^2|.
The first formula follows from the first Cloitre formula and a Liouville formula; the second formula follows from our analogous formula (cf. our comment in Formula section of A000005). (End)
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(tau(k)/k)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 23 2018

cp's OEIS Frontend

A005361 Product of exponents of prime factorization of n.

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A047994 Unitary totient (or unitary phi) function uphi(n).

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A023900 Dirichlet inverse of Euler totient function (A000010).

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A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

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A108951 Primorial inflation of n: Fully multiplicative with a(p) = p# for prime p, where x# is the primorial A034386(x).

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