A078429 -id:A078429 - cp's OEIS frontend

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

1, 1, 2, 3, 4, 2, 6, 5, 7, 4, 10, 6, 12, 6, 8, 11, 16, 7, 18, 12, 12, 10, 22, 10, 21, 12, 20, 18, 28, 8, 30, 21, 20, 16, 24, 21, 36, 18, 24, 20, 40, 12, 42, 30, 28, 22, 46, 22, 43, 21, 32, 36, 52, 20, 40, 30, 36, 28, 58, 24, 60, 30, 42, 43, 48, 20, 66, 48, 44, 24, 70, 35

Offset: 1

N. J. A. Sloane, Feb 06 2012

For more information see the Comments in A061020.
a(n) is the number of integers j such that 1 <= j <= n and gcd(n,j) is a perfect square. For example, a(12) = 6 because |{1,4,5,7,8,11}|=6 and the respective GCDs with 12 are 1,4,1,1,4,1, which are squares. - Geoffrey Critzer, Feb 16 2015
If m is squarefree (A005117), then a(m) = A000010(m) where A000010 is the Euler totient function. - Michel Marcus, Nov 08 2017
Also it appears that the primorials (A002110) is the sequence of indices of minimum records for a(n)/n, and these records are A038110(n)/A060753(n). - Michel Marcus, Nov 09 2017
Also called rho(n). When rho(n) | n, then n is called k-imperfect, with k = n/rho(n), cf. A127724. - M. F. Hasler, Feb 13 2020

P. J. McCarthy, Introduction to Arithmetical Functions, Springer Verlag, 1986, page 25.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
László Tóth, A survey of the alternating sum-of-divisors function, arXiv:1111.4842 [math.NT], 2011-2014.

Cf. A061020, A206368.
Cf. A027748 row, A124010, A206475 (first differences).
Cf. A078429.
Cf. A127724 (k-imperfect), A127725 (2-imperfect), A127726 (3-imperfect).

a206369 n = product $
   zipWith h (a027748_row n) (map toInteger $ a124010_row n) where
           h p e = sum $ take (fromInteger e + 1) $
                         iterate ((* p) . negate) (1 - 2 * (e `mod` 2))
-- Reinhard Zumkeller, Feb 08 2012

a:= n-> mul(add(i[1]^(i[2]-j)*(-1)^j, j=0..i[2]), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Nov 03 2017

Table[Length[Select[Range[n], IntegerQ[GCD[n, #]^(1/2)] &]], {n, 72}] (* Geoffrey Critzer, Feb 16 2015 *)
a[n_] := n*DivisorSum[n, LiouvilleLambda[#]/#&]; Array[a, 72] (* Jean-François Alcover, Dec 04 2017, after Enrique Pérez Herrero *)
f[p_,e_] := Sum[(-1)^(e-k)*p^k, {k,0,e}]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jan 01 2020 *)

a(n) = sum(k=1, n, issquare(gcd(n, k)));

ak(p,e)=my(s=1); for(i=1,e, s=s*p + (-1)^i); s
a(n)=my(f=factor(n)); prod(i=1,#f~, ak(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Dec 27 2016

a(n) = sumdiv(n, d, eulerphi(n/d) * issquare(d)); \\ Daniel Suteu, Jun 27 2018

apply( {A206369(n)=vecprod([f[1]^(f[2]+1)\/(f[1]+1)|f<-factor(n)~])}, [1..99]) \\ M. F. Hasler, Feb 13 2020

from math import prod
from sympy import factorint
def A206369(n): return prod((lambda x:x[0]+int((x[1]<<1)>=p+1))(divmod(p**(e+1),p+1)) for p, e in factorint(n).items()) # Chai Wah Wu, Mar 05 2024

a(n) = abs(A061020(n)).
a(n) = n*Sum_{d|n} lambda(d)/d, where lambda(n) is A008836(n). - Enrique Pérez Herrero, Sep 23 2012
Dirichlet g.f.: zeta(s - 1)*zeta(2*s)/zeta(s). - Geoffrey Critzer, Feb 25 2015
From Michel Marcus, Nov 05 2017: (Start)
a(2^n) = A001045(n+1);
a(3^n) = A015518(n+1);
a(5^n) = A015531(n+1);
a(7^n) = A015552(n+1);
a(11^n) = A015592(n+1). (End)
a(p^k) = p^k - a(p^(k - 1)) for k > 0 and prime p. - David A. Corneth, Nov 09 2017
a(n) = Sum_{d|n, d is a perfect square} phi(n/d), where phi(k) is the Euler totient function. - Daniel Suteu, Jun 27 2018
a(p^k) = A071324(p^k), for k >= 0 and prime p. - Michel Marcus, Aug 11 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / 30. - Vaclav Kotesovec, Feb 07 2019
G.f.: Sum_{k>=1} lambda(k)*x^k/(1 - x^k)^2. - Ilya Gutkovskiy, May 23 2019
a(n) = Sum_{i=1..n} A010052(gcd(n,i)). - Ridouane Oudra, Nov 24 2019
a(p^k) = round(p^(k+1)/(p+1)). - M. F. Hasler, Feb 13 2020

A384053 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cube.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 8, 8, 4, 10, 6, 12, 6, 8, 15, 16, 8, 18, 12, 12, 10, 22, 16, 24, 12, 27, 18, 28, 8, 30, 31, 20, 16, 24, 24, 36, 18, 24, 32, 40, 12, 42, 30, 32, 22, 46, 30, 48, 24, 32, 36, 52, 27, 40, 48, 36, 28, 58, 24, 60, 30, 48, 64, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

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

Unitary analog of A078429.
Cf. A013664, A384046, A384047.
The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is: A047994 (1), A384048 (squarefree), A384049 (cubefree), A384050 (powerful), A384051 (cubefull), A384052 (square), this sequence (cube), A384054 (exponentially odd), A384055 (odd), A384056 (power of 2), A384057 (3-smooth), A384058 (5-rough).

f[p_, e_] := p^e - If[Divisible[e, 3], 0, 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]^f[i,2] - if(f[i,2]%3, 1, 0));}

Multiplicative with a(p^e) = p^e if e is a multiple of 3, and p^e-1 otherwise.
Dirichlet g.f.: zeta(s-1) * zeta(3*s) * Product_{p prime} (1 - 1/p^s - 1/p^(2*s) + 1/p^(2*s-1) - 1/p^(3*s) + 1/p^(3*s-1)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(6) * Product_{p prime} (1 - 1/p^2 + 1/p^3 - 1/p^4 + 1/p^5 - 1/p^6) = 0.71190515701599590826... .

A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 9, 10, 11, 12, 13, 14, 15, 18, 17, 18, 19, 20, 21, 22, 23, 27, 25, 26, 28, 28, 29, 30, 31, 36, 33, 34, 35, 36, 37, 38, 39, 45, 41, 42, 43, 44, 45, 46, 47, 54, 49, 50, 51, 52, 53, 56, 55, 63, 57, 58, 59, 60, 61, 62, 63, 73, 65, 66, 67, 68, 69, 70, 71, 81, 73, 74, 75

Offset: 1

Ilya Gutkovskiy, Sep 19 2019

Sum of divisors d of n such that n/d is a cube.

Inverse Moebius transform of A078429.

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

Cf. A000578, A004709 (fixed points), A010057, A061704, A076752, A078429, A113061.

nmax = 75; CoefficientList[Series[Sum[x^(k^3)/(1 - x^(k^3))^2, {k, 1, Floor[nmax^(1/3)] + 1}], {x, 0, nmax}], x] // Rest
a[n_] := DivisorSum[n, # &, IntegerQ[(n/#)^(1/3)] &]; Table[a[n], {n, 1, 75}]
f[p_, e_] := (p^(e+3) - p^Mod[e, 3])/(p^3-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 25 2025 *)

A327626(n) = sumdiv(n,d,ispower(n/d,3)*d); \\ Antti Karttunen, Sep 19 2019

a(n) = Sum_{d|n} A078429(d).
a(n) = Sum_{d|n} A010057(n/d) * d. Dirichlet convolution of A000027 and A010057.
D.g.f.: zeta(s-1)*zeta(3s). - R. J. Mathar, Jun 05 2020
Sum_{k=1..n} a(k) ~ Pi^6 * n^2 / 1890. - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = (p^(e+3) - p^(e mod 3))/(p^3-1). - Amiram Eldar, May 25 2025

A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

Original entry on oeis.org

1, 1, 2, 3, 4, 2, 6, 6, 7, 4, 10, 6, 12, 6, 8, 12, 16, 7, 18, 12, 12, 10, 22, 12, 21, 12, 21, 18, 28, 8, 30, 24, 20, 16, 24, 21, 36, 18, 24, 24, 40, 12, 42, 30, 28, 22, 46, 24, 43, 21, 32, 36, 52, 21, 40, 36, 36, 28, 58, 24, 60, 30, 42, 48, 48, 20, 66, 48, 44

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the powerfree part (A055231) of gcd(n,k) is 1.

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

Cf. A000010, A001694, A005117, A050873, A055231, A063658, A206369.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), this sequence (powerful), A384040 (cubefull), A384041 (exponentially odd), A384042 (5-rough).

f[p_, e_] := If[e == 1, p-1, (p^2-p+1)*p^(e-2)]; 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, f[i,1]-1, (f[i,1]^2-f[i,1]+1)*f[i,1]^(f[i,2]-2)));}

Multiplicative with a(p^e) = (p^2-p+1)*p^(e-2) if e >= 2, and p-1 otherwise.
a(n) >= A000010(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(2*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^4) = 0.66922021803510257394... .

A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 6, 5, 6, 4, 10, 4, 12, 6, 8, 10, 16, 6, 18, 8, 12, 10, 22, 10, 20, 12, 19, 12, 28, 8, 30, 20, 20, 16, 24, 12, 36, 18, 24, 20, 40, 12, 42, 20, 24, 22, 46, 20, 42, 20, 32, 24, 52, 19, 40, 30, 36, 28, 58, 16, 60, 30, 36, 40, 48, 20, 66, 32, 44, 24

Offset: 1

Amiram Eldar, May 18 2025

The number of integers k from 1 to n such that the cubefree part (A360539) of gcd(n,k) is 1.

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

Cf. A005117, A036966, A078429, A360539, A254926.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), this sequence (cubefull), A384041 (exponentially odd), A384042 (5-rough).

f[p_, e_] := Switch[e, 1, p-1, 2, p^2-p, , (p^3-p^2+1)*p^(e-3)]; 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, f[i,1]-1, if(f[i,2] == 2, f[i,1]*(f[i,1]-1), (f[i,1]^3-f[i,1]^2+1)*f[i,1]^(f[i,2]-3))));}

Multiplicative with a(p^e) = (p^3-p^2+1)*p^(e-3) if e >= 3, p*(p-1) if e = 2, and p-1 otherwise.
a(n) >= A384039(n), with equality if and only if n is squarefree (A005117).
Dirichlet g.f.: zeta(s-1) * Product_{p prime} (1 - 1/p^s + 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/p^6) = 0.62159731307414305346... .

A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

Original entry on oeis.org

1, 2, 3, 3, 5, 6, 7, 7, 8, 10, 11, 9, 13, 14, 15, 13, 17, 16, 19, 15, 21, 22, 23, 21, 24, 26, 25, 21, 29, 30, 31, 27, 33, 34, 35, 24, 37, 38, 39, 35, 41, 42, 43, 33, 40, 46, 47, 39, 48, 48, 51, 39, 53, 50, 55, 49, 57, 58, 59, 45, 61, 62, 56, 53, 65, 66, 67, 51

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A268335.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), this sequence (exponentially odd), A384042 (5-rough).

f[p_, e_] := ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+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,1]-1)*f[i,1]^(f[i,2]-1) - (-1)^f[i,2])/(f[i,1] + 1));}

Multiplicative with a(p^e) = ((p^2+p-1)*p^(e-1) - (-1)^e)/(p+1).
a(n) >= A000010(n), with equality if and only if n = 1.
Dirichlet g.f.: (zeta(s-1)*zeta(2*s)/zeta(s)) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 - 1/p^2 + 1/(p^2+1)) = 0.93749428273130025078... .

A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

Original entry on oeis.org

1, 1, 2, 2, 5, 2, 7, 4, 6, 5, 11, 4, 13, 7, 10, 8, 17, 6, 19, 10, 14, 11, 23, 8, 25, 13, 18, 14, 29, 10, 31, 16, 22, 17, 35, 12, 37, 19, 26, 20, 41, 14, 43, 22, 30, 23, 47, 16, 49, 25, 34, 26, 53, 18, 55, 28, 38, 29, 59, 20, 61, 31, 42, 32, 65, 22, 67, 34, 46

Offset: 1

Amiram Eldar, May 18 2025

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

Cf. A000010, A003586, A007310, A065330, A065331, A372671.

The number of integers k from 1 to n such that gcd(n,k) is: A026741 (odd), A062570 (power of 2), A063659 (squarefree), A078429 (cube), A116512 (power of a prime), A117494 (prime), A126246 (1 or 2), A206369 (square), A254926 (cubefree), A372671 (3-smooth), A384039 (powerful), A384040 (cubefull), A384041 (exponentially odd), this sequence (5-rough).

f[p_, e_] := If[p < 5, (p-1)*p^(e-1), p^e]; 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,1] < 5, (f[i,1]-1)*f[i,1]^(f[i,2]-1), f[i,1]^f[i,2]));}

Multiplicative with a(p^e) = (p-1)*p^(e-1) if p <= 3 and p^e if p >= 5.
a(n) >= A000010(n), with equality if and only if n is 3-smooth (A003586).
a(n) = A000010(A065331(n)) * A065330(n).
a(n) = 2 * n * phi(n)/phi(6*n) = n * A000010(n) / A372671(n).
Dirichlet g.f.: zeta(s-1) * (1-1/2^s) * (1-1/3^s).
Sum_{k=1..n} a(k) ~ n^2 / 3.

cp's OEIS Frontend

A206369 a(p^k) = p^k - p^(k-1) + p^(k-2) - ... +- 1, and then extend by multiplicativity.

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A384053 The number of integers k from 1 to n such that the greatest divisor of k that is a unitary divisor of n is a cube.

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A327626 Expansion of Sum_{k>=1} x^(k^3) / (1 - x^(k^3))^2.

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A384039 The number of integers k from 1 to n such that gcd(n,k) is a powerful number.

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A384040 The number of integers k from 1 to n such that gcd(n,k) is a cubefull number.

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A384041 The number of integers k from 1 to n such that gcd(n,k) is an exponentially odd number.

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A384042 The number of integers k from 1 to n such that gcd(n,k) is a 5-rough number (A007310).

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