A295295 -id:A295295 - cp's OEIS frontend

A092261 Sum of unitary, squarefree divisors of n, including 1.

1, 3, 4, 1, 6, 12, 8, 1, 1, 18, 12, 4, 14, 24, 24, 1, 18, 3, 20, 6, 32, 36, 24, 4, 1, 42, 1, 8, 30, 72, 32, 1, 48, 54, 48, 1, 38, 60, 56, 6, 42, 96, 44, 12, 6, 72, 48, 4, 1, 3, 72, 14, 54, 3, 72, 8, 80, 90, 60, 24, 62, 96, 8, 1, 84, 144, 68, 18, 96, 144, 72, 1, 74, 114, 4, 20, 96, 168, 80

Offset: 1

Steven Finch, Feb 20 2004

Unitary convolution of the sequence of n*mu^2(n) (absolute values of A055615) and A000012. - R. J. Mathar, May 30 2011

Michael De Vlieger, 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, sequence sigma'(n).
Steven R. Finch, Unitarism and Infinitarism, February 25, 2004. [Cached copy, with permission of the author]

Cf. A000203, A003557, A007947, A048250, A055231, A056671, A057521, A065465, A295294, A295295, A329728.

Table[Plus @@ Select[Divisors@ n, Max @@ Last /@ FactorInteger@ # == 1 && GCD[#, n/#] == 1 &], {n, 1, 79}] (* Michael De Vlieger, Mar 08 2015 *)
f[p_, e_] := If[e==1, p+1, 1]; a[1]=1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 79] (* Amiram Eldar, Mar 01 2019 *)

a(n) = sumdiv(n, d, d*issquarefree(d)*(gcd(d, n/d) == 1)); \\ Michel Marcus, Mar 06 2015

for(n=1, 100, print1(direuler(p=2, n, (1 + p^2*X^3 - p*X^2 - p^2*X^2)/(1-X)/(1-p*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

;; This implementation utilizes the memoization-macro definec for which an implementation is available at http://oeis.org/wiki/Memoization#Scheme
;; The other functions, A020639, A067029 and A028234 can be found under the respective entries, and should likewise defined with definec:
(definec (A092261 n) (if (= 1 n) 1 (* (+ 1 (if (> (A067029 n) 1) 0 (A020639 n))) (A092261 (A028234 n))))) ;; Antti Karttunen, Nov 25 2017

Multiplicative with a(p) = p+1 and a(p^e) = 1 for e > 1. - Vladeta Jovovic, Feb 22 2004
From Álvar Ibeas, Mar 06 2015: (Start)
a(n) = a(A055231(n)) = A000203(A055231(n)).
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + p^(1-s) - p^(1-2s)).
(End)
From Antti Karttunen, Nov 25 2017: (Start)
a(n) = A048250(A055231(n)).
a(n) = A000203(n) / A295294(n).
a(n) = A048250(n) / A295295(n) = A048250(n) / A048250(A057521(n)), where A057521(n) = A064549(A003557(n)).
(End)
Lim_{n->oo} (1/n) * Sum_{k=1..n} a(k)/k = Product_{p prime}(1 - 1/(p^2*(p+1))) = 0.881513... (A065465). - Amiram Eldar, Jun 10 2020
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 + p^(2-3*s) - p^(1-2*s) - p^(2-2*s)). - Vaclav Kotesovec, Aug 20 2021
a(n) = Sum_{d|n, gcd(d,n/d)=1} d * mu(d)^2. - Wesley Ivan Hurt, May 26 2023

A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

Original entry on oeis.org

1, 1, 1, 7, 1, 1, 1, 15, 13, 1, 1, 7, 1, 1, 1, 31, 1, 13, 1, 7, 1, 1, 1, 15, 31, 1, 40, 7, 1, 1, 1, 63, 1, 1, 1, 91, 1, 1, 1, 15, 1, 1, 1, 7, 13, 1, 1, 31, 57, 31, 1, 7, 1, 40, 1, 15, 1, 1, 1, 7, 1, 1, 13, 127, 1, 1, 1, 7, 1, 1, 1, 195, 1, 1, 31, 7, 1, 1, 1, 31, 121, 1, 1, 7, 1, 1, 1, 15, 1, 13, 1, 7, 1, 1, 1, 63

Offset: 1

Antti Karttunen, Nov 25 2017

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences related to sums of divisors.

Cf. A000203, A001694, A003557, A005117, A057521, A064549, A092261, A295295.

Array[DivisorSigma[1, #/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] ] &, 96] (* Michael De Vlieger, Nov 26 2017, after Jean-François Alcover at A057521 *)
f[p_, e_] := If[e == 1, 1, (p^(e+1)-1)/(p-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 08 2022 *)

a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i=1, #p, if(e[i] == 1, 1, (p[i]^(e[i]+1)-1)/(p[i]-1)))}; \\ Amiram Eldar, Oct 08 2022

from math import prod
from sympy import factorint
def A295294(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if e > 1) # Chai Wah Wu, Nov 14 2022

(define (A295294 n) (A000203 (A057521 n)))
;; With memoization-macro definec:
(definec (A295294 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (if (= e 1) 1 (/ (- (expt p (+ 1 e)) 1) (- p 1))) (A295294 (A028234 n))))))

Multiplicative with a(p) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e > 1.
a(n) = A000203(A057521(n)) = A000203(A064549(A003557(n))).
a(n) = A000203(n) / A092261(n).
From Amiram Eldar, Oct 08 2022: (Start)
a(n) = 1 iff n is squarefree (A005117).
a(n) = A000203(n) iff n is powerful (A001694). (End)
Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(3*s-2)). - Amiram Eldar, Sep 09 2023

A344137 Sum of the squarefree divisors of n whose square does not divide n.

Original entry on oeis.org

0, 2, 3, 0, 5, 11, 7, 0, 0, 17, 11, 9, 13, 23, 23, 0, 17, 8, 19, 15, 31, 35, 23, 9, 0, 41, 0, 21, 29, 71, 31, 0, 47, 53, 47, 0, 37, 59, 55, 15, 41, 95, 43, 33, 20, 71, 47, 9, 0, 12, 71, 39, 53, 8, 71, 21, 79, 89, 59, 69, 61, 95, 28, 0, 83, 143, 67, 51, 95, 143, 71, 0, 73

Offset: 1

Wesley Ivan Hurt, Jun 16 2021

			a(20) = Sum_{d|20} d * mu(d)^2 * c(20/d^2) = 1*1*0 + 2*1*0 + 4*0*1 + 5*1*1 + 10*1*1 + 20*0*1 = 15.

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

Cf. A048250, A295295, A345320.

a[n_] := DivisorSum[n, # &, SquareFreeQ[#] && ! Divisible[n, #^2] &]; Array[a, 100] (* Amiram Eldar, Oct 13 2023 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i, 1] + 1) - prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1] + 1));} \\ Amiram Eldar, Oct 13 2023

a(n) = Sum_{d|n} d * mu(d)^2 * c(n/d^2), where c(n) = ceiling(n) - floor(n).

a(n) = A048250(n) - A295295(n). - Amiram Eldar, Oct 13 2023

A365336 The sum of exponentially odd divisors of the square root of the largest square dividing n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 11, 1, 1, 1, 3, 1, 1, 1, 12, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1

Offset: 1

Amiram Eldar, Sep 01 2023

First differs from A295295 at n = 64.
The sum of divisors of the square root of the largest square dividing n is A069290(n).
The number of these divisors is A365335(n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms)

Cf. A000188, A005117, A033634, A069290, A295295, A365335.

f[p_, e_] := (p^(2*Floor[(e+2)/4] + 1) - p)/(p^2 - 1) + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i,1]^(2*((f[i,2]+2)\4) + 1) - f[i,1])/(f[i,1]^2 - 1) + 1);}

a(n) = A033634(A000188(n)).
a(n) = 1 if and only if n is squarefree (A005117).
Multiplicative with a(p^e) = (p^(2*floor((e+2)/4) + 1) - p)/(p^2 - 1) + 1. [corrected by Georg Fischer, Oct 07 2023]
Dirichlet g.f.: zeta(s) * zeta(4*s-2) * Product_{p prime} (1 + 1/p^(2*s-1) - 1/p^(4*s-2)).
From Vaclav Kotesovec, Sep 02 2023: (Start)
Dirichlet g.f.: zeta(s)^2 * zeta(4*s-2) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s-1) - 1/p^(3*s-1) - 1/p^(4*s-2) + 1/p^(5*s-2)).
Dirichlet g.f.: zeta(s) * zeta(2*s-1) * zeta(4*s-2) * Product_{p prime} (1 - 2/p^(4*s-2) + 1/p^(6*s-3)).
Let f(s) = Product_{p prime} (1 - 2/p^(4*s-2) + 1/p^(6*s-3)), then Sum_{k=1..n} a(k) ~ Pi^2/12 * n * (f(1) * (log(n) + 3*gamma - 1 + 24*zeta'(2)/Pi^2) + f'(1)), where f(1) = Product_{p prime} (1 - 2/p^2 + 1/p^3) = A065464 = 0.42824950567709444021876..., f'(1) = f(1) * Sum_{primes p} 2*(4*p-3)*log(p) /  (p^3 - 2*p + 1) = 1.617322217899181826790... and gamma is the Euler-Mascheroni constant A001620. (End)

A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 6, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 3, 4, 1, 1, 3, 8, 6, 1, 3, 1, 4, 1, 3, 1, 1, 1, 3, 1, 1, 4, 3, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 1, 3, 4, 1, 1, 3, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 02 2024

			Fractions begin with: 1, 1, 1, 3/2, 1, 1, 1, 3/2, 4/3, 1, 1, 3/2, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Cf. A005117, A057521, A157289, A295295, A332880, A370784 (denominators).

a[n_] := Numerator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]

a(n) = {my(f = factor(n)); numerator(prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + 1/f[i,1])));}

a(n) = A332880(A057521(n)).
Let f(n) = a(n)/A370784(n):
f(n) is multiplicative with f(p) = 1 and f(p^e) = 1 + 1/p for e >= 2.
f(n) = 1 if and only if n is squarefree (A005117).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} f(k) = zeta(3)/zeta(6) = 1.181564... (A157289) (Jakimczuk, 2024).

A370784 a(n) is the denominator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 5, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 7, 5, 1, 2, 1, 3, 1, 2, 1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 5, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1

Offset: 1

Amiram Eldar, Mar 02 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Rafael Jakimczuk, Arithmetical Functions over the Powerful Part of an Integer, ResearchGate, 2024.

Cf. A005117, A057521, A295295, A332881, A370783 (numerators).

a[n_] := Denominator[Times @@ (1 + 1/Select[FactorInteger[n], Last[#] > 1 &][[;; , 1]])]; Array[a, 100]

a(n) = {my(f = factor(n)); denominator(prod(i = 1, #f~, if(f[i,2] == 1, 1, 1 + 1/f[i,1])));}

a(n) = A332881(A057521(n)).

a(n) = 1 if n is squarefree (A005117).

A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 4, 6, 1, 1, 2, 1, 1, 1, 8, 1, 6, 1, 2, 1, 1, 1, 4, 20, 1, 18, 2, 1, 1, 1, 16, 1, 1, 1, 12, 1, 1, 1, 4, 1, 1, 1, 2, 6, 1, 1, 8, 42, 20, 1, 2, 1, 18, 1, 4, 1, 1, 1, 2, 1, 1, 6, 32, 1, 1, 1, 2, 1, 1, 1, 24, 1, 1, 20, 2, 1, 1, 1, 8, 54, 1, 1, 2

Offset: 1

Amiram Eldar, Jan 13 2025

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

Cf. A000010, A001694, A005117, A057521, A295294, A295295, A323333, A349379, A357669, A380161.

f[p_, e_] := If[e == 1, 1, (p-1)*p^(e-1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2] == 1, 1, (f[i, 1]-1)*f[i, 1]^(f[i, 2]-1)));}

a(n) = A000010(A057521(n)).
a(n) >= 1, with equality if and only if n is squarefree (A005117).
a(n) <= A000010(n), with equality if and only if n is powerful (A001694).
Multiplicative with a(p) = 1, and a(p^e) = (p-1)*p^(e-1) if e >= 2.
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 + 1/p^s - 1/p^(s-1) + 1/p^(2*s-2) - 2/p^(2*s-1)).
Sum_{k=1..n} a(k) ~ c * n^(3/2) / 3, where c = Product_{p prime} (1 + 2/p^(3/2) - 1/p^2 - 2/p^(5/2)) = 1.96428740396979919886... .

cp's OEIS Frontend

A092261 Sum of unitary, squarefree divisors of n, including 1.

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A295294 Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).

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A344137 Sum of the squarefree divisors of n whose square does not divide n.

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A365336 The sum of exponentially odd divisors of the square root of the largest square dividing n.

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A370783 a(n) is the numerator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

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A370784 a(n) is the denominator of the sum of the reciprocals of the squarefree divisors of the powerful part of n.

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A380160 a(n) is the value of the Euler totient function when applied to the powerful part of n.

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