A192066 -id:A192066 - cp's OEIS frontend

A206787 Sum of the odd squarefree divisors of n.

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

Offset: 1

Reinhard Zumkeller, Feb 12 2012

a(A000079(n)) = 1; a(A057716(n)) > 1; a(A065119(n)) = 4; a(A086761(n)) = 6.

Inverse Mobius transform of 1, 0, 3, 0, 5, 0, 7, 0, 0, 0, 11, 0, 13, 0, 15, 0, 17, 0, 19, 0, 21, 0, 23, 0, 0, 0, 0, 0, 29... - R. J. Mathar, Jul 12 2012

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Jon Maiga, Computer-generated formulas for A206787, Sequence Machine.

Cf. A000203, A000593, A007947, A008683, A056911, A048250, A010684, A192066, A204455.
Cf. A000079, A057716, A065119, A086761.
Inverse Möbius transform of the absolute values of A349343.

a206787 = sum . filter odd . a206778_row

[&+[d:d in Divisors(m)|IsOdd(d) and IsSquarefree(d)]:m in [1..72]]; // Marius A. Burtea, Aug 14 2019

seq(add(d*mobius(2*d)^2, d in divisors(n)), n=1 .. 80); # Ridouane Oudra, Aug 14 2019

a[n_] := DivisorSum[n, #*Boole[OddQ[#] && SquareFreeQ[#]]&]; Array[a, 80] (* Jean-François Alcover, Dec 05 2015 *)
f[2, e_] := 1; f[p_, e_] := p + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

a(n) = sumdiv(n, d, d*(d % 2)*issquarefree(d)); \\ Michel Marcus, Sep 21 2014

from math import prod
from sympy import primefactors
def A206787(n): return prod(1+(p if p>2 else 0) for p in primefactors(n)) # Chai Wah Wu, Oct 10 2024

a(n) = Sum_{k = 1..A034444(n)} (A206778(n,k) mod 2) * A206778(n,k).
a(n) = Sum_{d|n} d*mu(2*d)^2, where mu is the Möbius function (A008683). - Ridouane Oudra, Aug 14 2019
Multiplicative with a(2^e) = 1, and a(p^e) = p + 1 for p > 2. - Amiram Eldar, Sep 18 2020
Sum_{k=1..n} a(k) ~ (1/3) * n^2. - Amiram Eldar, Nov 17 2022
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-2))*(2^s/(2^s+2)). - Amiram Eldar, Jan 03 2023
From Antti Karttunen, Nov 22 2023: (Start)
a(n) = A000203(A204455(n)) = A000593(A007947(n)) = A048250(n)/A010684(n-1). [From Sequence Machine]
a(n) = Sum_{d|n} abs(A349343(d)). [See R. J. Mathar's Jul 12 2012 comment above]
(End)
a(n) = Sum_{d divides n, d odd} d * mu(d)^2. - Peter Bala, Feb 01 2024

A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 1, 10, 18, 12, 20, 14, 24, 24, 17, 18, 30, 20, 30, 32, 36, 24, 4, 26, 42, 1, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 6, 42, 96, 44, 60, 60, 72, 48, 68, 50, 78, 72, 70, 54, 3, 72, 8, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90, 96

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A385042(n), and the largest of them is A367168(n).

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

The unitary analog of A353900.
Cf. A005117, A034448, A138302, A209229, A385042.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), this sequence (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[e == 2^IntegerExponent[e, 2], p^e + 1, 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<

Multiplicative with a(p^e) = p^(A209229(e)) + 1.
a(n) <= A034448(n), with equality if and only if n is in A138302.
a(n) <= A353900(n), with equality if and only if n is squarefree (A005117).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1/(p*(p+1)) + Sum_{k>=2} (1/p^(2^k)-1/p^(2^k-1))) = 1.21427559551509410114... .

A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 8, 1, 1, 6, 12, 1, 14, 8, 6, 1, 18, 1, 20, 6, 8, 12, 24, 1, 26, 14, 1, 8, 30, 6, 32, 1, 12, 18, 48, 1, 38, 20, 14, 6, 42, 8, 44, 12, 6, 24, 48, 1, 50, 26, 18, 14, 54, 1, 72, 8, 20, 30, 60, 6, 62, 32, 8, 1, 84, 12, 68, 18, 24, 48, 72, 1, 74, 38

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A186099 at n = 25; a(25) = 26, while A186099(25) = 31.

The number of these divisors is A385044(n), and the largest of them is A065330(n).

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

The unitary analog of A186099.
Cf. A002117, A007310, A034448, A065330, A385044.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), this sequence (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[p <= 3, 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,1] < 5, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if p <= 3, and p^e + 1 if p >= 5.
a(n) = A034448(n)/A385046(n).
a(n) <= A034448(n), with equality if and only if n is 5-rough.
a(n) <= A186099(n).
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1)) * ((1-1/2^(s-1))/(1-1/2^(2*s-1))) * ((1-1/3^(s-1))/(1-1/3^(2*s-1))).
Sum_{k=1..n} a(k) ~ c * n^2, where c = 3*Pi^2/(91*zeta(3)) = 0.270679... .

A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

Original entry on oeis.org

1, 3, 4, 5, 1, 12, 1, 9, 10, 3, 1, 20, 1, 3, 4, 17, 1, 30, 1, 5, 4, 3, 1, 36, 1, 3, 28, 5, 1, 12, 1, 33, 4, 3, 1, 50, 1, 3, 4, 9, 1, 12, 1, 5, 10, 3, 1, 68, 1, 3, 4, 5, 1, 84, 1, 9, 4, 3, 1, 20, 1, 3, 10, 65, 1, 12, 1, 5, 4, 3, 1, 90, 1, 3, 4, 5, 1, 12, 1, 17

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A382488(n), and the largest of them is A065331(n).

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

The unitary analog of A072079.
Cf. A001620, A003586, A034448, A065331, A082633, A382488.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), this sequence (3-smooth), A385047 (power of 2), A385048 (cubefull), A385049 (biquadratefree).

f[n_, p_] := If[Divisible[n, p], p^IntegerExponent[n, p] + 1, 1]; a[n_] := f[n, 2]*f[n, 3]; Array[a, 100]

a(n) = if(n%2, 1, 2^valuation(n, 2)+1) * if(!(n%3), 3^valuation(n, 3)+1, 1);

Multiplicative with a(p^e) = p^e + 1 if p <= 3, and 1 if p >= 5.
a(n) = A034448(n)/A385045(n).
a(n) <= A034448(n), with equality if and only if n 3-smooth.
a(n) <= A072079(n).
Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))) * ((1-1/3^(2*s-1))/(1-1/3^(s-1))).
Sum_{k=1..n} a(k) ~ (n/(6*log(2)*log(3))) * (log(n)^2 + c1*log(n) + c2), where c1 = 2*gamma - 2 + 7*log(2) + 5*log(3) - 2*log(6) = 5.916004..., c2 = 2 - 5*log(2) - 11*log(2)^2/6 - 3*log(3) - 5*log(3)^2/6 + 15*log(2)*log(3)/2 + (5*log(2) + 3*log(3) - 2)*gamma - 2*gamma_1 = 1.957142..., gamma is Euler's constant (A001620), and gamma_1 is the 1st Stieltjes constant (A082633).

A385047 The sum of the unitary divisors of n that are powers of 2.

Original entry on oeis.org

1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 33, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 65, 1, 3, 1, 5, 1, 3, 1, 9, 1, 3, 1, 5, 1, 3, 1, 17, 1, 3, 1, 5, 1, 3

Offset: 1

Amiram Eldar, Jun 16 2025

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

The unitary analog of A038712.
Cf. A001620, A006519, A034448.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), this sequence (power of 2), A385048 (cubefull), A385049 (biquadratefree).

a[n_] := If[OddQ[n], 1, 2^IntegerExponent[n, 2] + 1]; Array[a, 100]

PARI
```
a(n) = if(n%2, 1, 2^valuation(n, 2)+1);
```

Multiplicative with a(2^e) = 2^e + 1, and a(p^e) = 1 for an odd prime p.
a(n) = A034448(n) / A192066(n).
a(n) = A059841(n) + A006519(n), i.e., a(n) = A006519(n) + 1 if n is even, and 1 is n is odd.
Dirichlet g.f.: zeta(s) * ((1-1/2^(2*s-1))/(1-1/2^(s-1))).
Sum_{k=1..n} a(k) ~ (n/(2*log(2))) * (log(n) + gamma - 1 + 5*log(2)/2), where gamma is Euler's constant (A001620).

A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 28, 1, 1, 1, 1, 33, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 1, 1, 1, 1, 1, 28, 1, 9, 1, 1, 1, 1, 1, 1, 1, 65, 1, 1, 1, 1, 1, 1, 1, 9, 1, 1, 1, 1, 1, 1, 1, 17, 82, 1, 1, 1, 1

Offset: 1

Amiram Eldar, Jun 16 2025

The number of these divisors is A368248(n), and the largest of them is A360540(n).

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

The unitary analog of A385005.
Cf. A034448, A036966, A046100, A360540, A368248.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), this sequence (cubefull), A385049 (biquadratefree).

f[p_, e_] := If[e <= 2, 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] < 3, 1, f[i, 1]^f[i, 2] + 1));}

Multiplicative with a(p^e) = 1 if e <= 2, and a(p^e) = p^e + 1 if e >= 3.
a(n) = A034448(n) / A371242(n).
a(n) <= A034448(n), with equality if and only if n is cubefull (A036966).
a(n) <= A385005(n), with equality if and only if n is biquadratefree (A046100).
Dirichlet g.f.: zeta(s)*zeta(s-1)*Product_{p prime} (1 - 1/p^(s-1) + 1/p^(3*s-3) - 1/p^(4*s-3)).

A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

Original entry on oeis.org

1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, 1, 18, 30, 20, 30, 32, 36, 24, 36, 26, 42, 28, 40, 30, 72, 32, 1, 48, 54, 48, 50, 38, 60, 56, 54, 42, 96, 44, 60, 60, 72, 48, 4, 50, 78, 72, 70, 54, 84, 72, 72, 80, 90, 60, 120, 62, 96, 80, 1, 84, 144, 68, 90

Offset: 1

Amiram Eldar, Jun 16 2025

First differs from A383763 at n = 32.

The number of these divisors is A365499(n), and the largest of them is A385007(n).

D. Suryanarayana, The number and sum of k-free integers <= x which are prime to n, Indian J. Math., Vol. 11 (1969), pp. 131-139.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Francesco Pappalardi, A survey on k-freeness, Number Theory, Ramanujan Math. Soc. Lect. Notes Ser., Vol. 1 (2003), pp. 71-88.

The unitary analog of A385006.
Cf. A036967, A046100, A365499, A385007.
The sum of unitary divisors of n that are: A092261 (squarefree), A192066 (odd), A358346 (exponentially odd), A358347 (square), A360720 (powerful), A371242 (cubefree), A380396 (cube), A383763 (exponentially squarefree), A385043 (exponentially 2^n), A385045 (5-rough), A385046 (3-smooth), A385047 (power of 2), A385048 (cubefull), this sequence (biquadratefree).

f[p_, e_] := If[e < 4, p^e + 1, 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] < 4, f[i, 1]^f[i, 2] + 1, 1)); }

Multiplicative with a(p^e) = p^e + 1 for e <= 3, and a(p^e) = 1 for e >= 4.
a(n) = 1 if and only if n is 4-full (A036967).
a(n) <= A034448(n), with equality if and only if n is biquadratefree.
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^(s-1) + 1/p^(2*s-2) - 1/p^(2*s-1) + 1/p^(3*s-3) - 1/p^(3*s-2) - 1/p^(4*s-3)).
Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = Product_{p prime} (1 + 1/(p^2 + p) - 1/p^4) = 1.27769267395905900191... .

A328258 a(n) = Sum_{d|n, gcd(d,n/d) = 1} (-1)^(d + 1) * d.

Original entry on oeis.org

1, -1, 4, -3, 6, -4, 8, -7, 10, -6, 12, -12, 14, -8, 24, -15, 18, -10, 20, -18, 32, -12, 24, -28, 26, -14, 28, -24, 30, -24, 32, -31, 48, -18, 48, -30, 38, -20, 56, -42, 42, -32, 44, -36, 60, -24, 48, -60, 50, -26, 72, -42, 54, -28, 72, -56, 80, -30, 60, -72, 62, -32, 80, -63, 84

Offset: 1

Ilya Gutkovskiy, Oct 09 2019

Excess of sum of odd unitary divisors of n over sum of even unitary divisors of n.

a(n) = n+1 iff n is in A061345 \ {1}. - Bernard Schott, Mar 05 2023

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Unitary Divisor.

Cf. A002129, A034448, A077610, A109712, A192066, A254503, A306633, A360156.

[&+[(-1)^(d+1)*d:d in Divisors(n)|Gcd(d, n div d) eq 1]:n in [1..70]]; // Marius A. Burtea, Oct 10 2019

f:= proc(n) local t;
  mul(1 - (-1)^t[1] * t[1]^t[2], t=ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Oct 10 2019

a[n_] := Sum[Boole[GCD[d, n/d] == 1] (-1)^(d + 1) d, {d, Divisors[n]}]; Table[a[n], {n, 1, 65}]
a[1] = 1; a[n_] := Times @@ (1 - (-1)^First[#] First[#]^Last[#] & /@ FactorInteger[n]); Table[a[n], {n, 1, 65}]

a(n) = sumdiv(n, d, if (gcd(d,n/d) == 1, (-1)^(d + 1) * d)); \\ Michel Marcus, Oct 10 2019

If n = Product (p_j^k_j) then a(n) = Product (1 - (-1)^p_j * p_j^k_j).
If n odd, a(n) = usigma(n), where usigma = A034448.
Sum_{k=1..n} a(k) ~ c * n^2, where c = zeta(2)/(14*zeta(3)) = A306633 / 14 = 0.0977451... . - Amiram Eldar, Nov 17 2022
From Amiram Eldar, Jan 28 2023: (Start)
a(n) = 2 * A192066(n) - A034448(n).
a(n) = A192066(n) - A360156(n/2) if n is even, and A192066(n) otherwise.
Dirichlet g.f.: (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(2*s)-2^(s+2)+2)/(2^(2*s)-2). (End)

A360156 a(n) is the sum of the even unitary divisors of 2*n.

Original entry on oeis.org

2, 4, 8, 8, 12, 16, 16, 16, 20, 24, 24, 32, 28, 32, 48, 32, 36, 40, 40, 48, 64, 48, 48, 64, 52, 56, 56, 64, 60, 96, 64, 64, 96, 72, 96, 80, 76, 80, 112, 96, 84, 128, 88, 96, 120, 96, 96, 128, 100, 104, 144, 112, 108, 112, 144, 128, 160, 120, 120, 192, 124, 128

Offset: 1

Amiram Eldar, Jan 28 2023

a(n) is the unitary analog of A146076(2*n).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Octavio A. Agustín-Aquino, Wang-Sun Formula in GL(Z/2kZ), INTEGERS, Vol. 23 (2023), #A37; arXiv preprint, arXiv:2207.14495 [math.NT], 2022.

Cf. A002117, A034448, A100008, A146076, A171977, A192066, A328258.

usigma[n_] := Times @@ (1 + Power @@@ FactorInteger[n]); usigma[1] = 1; a[n_] := Module[{e = IntegerExponent[n, 2]}, 2^(e + 1) * usigma[n/2^e]]; Array[a, 100]

usigma(n) = {my(f = factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + 1)} ;
a(n) = {my(e = valuation(n, 2)); (1 << (e+1)) * usigma(n >> e); }

a(n) = Sum_{even d|(2*n), gcd(d, 2*n/d)=1} d.
a(n) = A034448(2*n) - A192066(2*n).
a(n) = A192066(2*n) - A328258(2*n).
a(n) = A171977(n) * A192066(n).
Sum_{k=1..n} a(k) ~ Pi^2 * n^2 / (7*zeta(3)).
Dirichlet g.f. of b(n): (zeta(s)*zeta(s-1)/zeta(2*s-1))*(2^(s+1)-2)/(2^(2*s)-2), where b(n) is the sum of the even unitary divisors of n: b(n) = a(n/2) if n is even and 0 otherwise.

cp's OEIS Frontend

A206787 Sum of the odd squarefree divisors of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

Python

Formula

A385043 The sum of the unitary divisors of n whose exponents in their prime factorizations are all powers of 2 (A138302).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385045 The sum of the unitary divisors of n that are 5-rough numbers (A007310).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385046 The sum of the unitary divisors of n that are 3-smooth numbers (A003586).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385047 The sum of the unitary divisors of n that are powers of 2.

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385048 The sum of the unitary divisors of n that are cubefull numbers (A036966).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A385049 The sum of the unitary divisors of n that are biquadratefree numbers (A046100).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Mathematica