A100008 -id:A100008 - cp's OEIS frontend

A100007 Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.

1, 2, 2, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 2, 2, 4, 4, 2, 4, 2, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 2, 4, 2, 2, 4, 4, 2, 2, 2, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 8, 2, 2, 4, 2, 4, 4, 4, 2, 4, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 4, 4, 2, 4, 4, 2, 8, 2, 2, 4, 2, 4, 4, 2, 2, 4, 4, 4, 4, 2, 2, 8, 2, 2, 4, 4, 4, 4, 4

Offset: 1

N. J. A. Sloane, Nov 20 2004

			a(13)=2 because among the three divisors of 25 only 1 and 25 are unitary.

Cf. A001620, A034444, A068068, A100008.

with(numtheory): for n from 1 to 120 do printf(`%d,`,2^nops(ifactors(2*n-1)[2])) od: # Emeric Deutsch, Dec 24 2004

a[n_] := 2^PrimeNu[2*n-1]; Array[a, 100] (* Amiram Eldar, Jan 28 2023 *)

a(n) = 2^omega(2*n-1); \\ Amiram Eldar, Jan 28 2023

From Ilya Gutkovskiy, Apr 28 2017: (Start)
a(n) = [x^(2*n-1)] Sum_{k>=1} mu(k)^2*x^k/(1 - x^k).
a(n) = 2^omega(2*n-1). (End)
From Amiram Eldar, Jan 28 2023: (Start)
a(n) = A034444(2*n-1) = A068068(2*n-1).
Sum_{k=1..n} a(k) ~ 4*n*((log(n) + 2*gamma - 1 + 7*log(2)/3 - 2*zeta'(2)/zeta(2)) / Pi^2, where gamma is Euler's constant (A001620). (End)

More terms from Emeric Deutsch, Dec 24 2004

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.

A346773 a(n) = Sum_{d|n} möbius(d)^n.

Original entry on oeis.org

1, 2, 0, 2, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 2, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 2, 0, 8, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 8, 0, 4, 0, 4, 0, 4, 0, 4, 0, 4, 0, 8, 0

Offset: 1

Seiichi Manyama, Aug 03 2021

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

Cf. A008683, A023887, A034444, A068068, A080323, A100008, A264782.

Table[Sum[MoebiusMu[d]^n,{d,Divisors[n]}],{n,103}] (* Stefano Spezia, Aug 03 2021 *)

PARI
```
a(n) = sumdiv(n, d, moebius(d)^n);
```
PARI
```
a(n) = if(n%2, 0^(n-1), 2^omega(2*n));
```

N=99; x='x+O('x^N); Vec(sum(k=1, N, (moebius(k)*x)^k/(1-(moebius(k)*x)^k)))

G.f.: Sum_{k>=1} (mu(k)*x)^k/(1 - (mu(k)*x)^k).

a(2*n-1) = 0^(n-1) and a(2*n) = A034444(2*n) = A100008(n) for n > 0.

cp's OEIS Frontend

A100007 Number of unitary divisors of 2n-1 (d such that d divides 2n-1, GCD(d,(2n-1)/d)=1). Bisection of A034444.

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

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A346773 a(n) = Sum_{d|n} möbius(d)^n.

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