A100371 -id:A100371 - cp's OEIS frontend

A226305 Numerator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5, 4095, 1, 17, 17, 65535, 1, 262143, 17, 65, 1, 4194303, 17, 69905, 1, 4161, 65, 268435455, 1, 1073741823, 257, 1025, 1, 53261, 13, 68719476735, 1, 4097, 257, 1099511627775, 1, 4398046511103, 1025, 3133, 1, 70368744177663, 257, 69810262081, 1, 65537, 4097

Offset: 1

N. J. A. Sloane, Jun 07 2013

			1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5/3, 4095, 1, 17/3, 17, 65535, 1, 262143, 17/3, 65/3, 1, 4194303, 17/5, 69905, 1, 4161, 65/3, 268435455, 1, 1073741823, 257, 1025/3, 1, 53261/3, 13, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
N. Bliss, B. Fulan, S. Lovett, and J. Sommars, Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials, Amer. Math. Monthly, 120 (2013), 519-536.

Cf. A226306, A100371.

f:=proc(a,M) local n,b,d,t1,t2;
b:=[];
for n from 1 to M do
t1:=divisors(n);
t2:=mul(a[d]^mobius(n/d), d in t1);
b:=[op(b),t2];
od;
b;
end;
a:=[seq(2^phi(n)-1,n=1..100)];
f(a,100);

Table[Numerator[Product[(2^EulerPhi[d] - 1)^MoebiusMu[n/d], {d, Divisors[n]}]], {n, 100}] (* Indranil Ghosh, Apr 14 2017 *)

from sympy import divisors, totient, mobius, prod
def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).numerator
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 14 2017

A226306 Denominator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 3, 1, 1, 1, 1, 3, 3, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 1, 3, 1, 3, 1, 1, 1, 3, 5, 1, 1, 1, 3, 1, 1, 1, 17, 1, 1, 3, 3, 1, 1, 3, 5, 3, 1, 1, 85, 1, 1, 7, 1, 15, 1, 1, 3, 3, 1, 1, 1, 1, 1, 1, 3, 3, 1, 1, 17, 1, 1, 1, 325, 15, 1, 3, 5, 1, 1, 21, 3, 3, 1, 3, 257, 1, 1, 1, 1

Offset: 1

N. J. A. Sloane, Jun 07 2013

			1, 1, 3, 3, 15, 1, 63, 5, 21, 1, 1023, 5/3, 4095, 1, 17/3, 17, 65535, 1, 262143, 17/3, 65/3, 1, 4194303, 17/5, 69905, 1, 4161, 65/3, 268435455, 1, 1073741823, 257, 1025/3, 1, 53261/3, 13, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..5000
N. Bliss, B. Fulan, S. Lovett, and J. Sommars, Strong Divisibility, Cyclotomic Polynomials, and Iterated Polynomials, Amer. Math. Monthly, 120 (2013), 519-536.

Cf. A100371, A226305.

Table[Denominator[Product[(2^EulerPhi[d] - 1)^MoebiusMu[n/d], {d, Divisors[n]}]], {n, 100}] (* Indranil Ghosh, Apr 14 2017 *)

from sympy import divisors, totient, mobius, prod
def a(n): return prod((2**totient(d) - 1)**mobius(n//d) for d in divisors(n)).denominator
print([a(n) for n in range(1, 51)]) # Indranil Ghosh, Apr 14 2017

A100587 Number of nonempty subsets of divisors of n.

Original entry on oeis.org

1, 3, 3, 7, 3, 15, 3, 15, 7, 15, 3, 63, 3, 15, 15, 31, 3, 63, 3, 63, 15, 15, 3, 255, 7, 15, 15, 63, 3, 255, 3, 63, 15, 15, 15, 511, 3, 15, 15, 255, 3, 255, 3, 63, 63, 15, 3, 1023, 7, 63, 15, 63, 3, 255, 15, 255, 15, 15, 3, 4095, 3, 15, 63, 127, 15, 255, 3, 63, 15, 255, 3, 4095, 3

Offset: 1

Labos Elemer, Dec 01 2004

A119347(n) <= a(n). - Reinhard Zumkeller, Jun 27 2015

			For all prime numbers p, a(p)=3, since those subsets are {{1,p},{1},{p}}.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Wasin So, Integral circulant graphs, Discr. Math. 306 (1) (2006) 153-158

Cf. A000005, A066781, A100371.

Cf. A119347.

a100587 = (subtract 1) . (2 ^) . a000005'
-- Reinhard Zumkeller, Jun 27 2015

A100587:=n->-1+2^numtheory[tau](n): seq(A100587(n), n=1..100); # Wesley Ivan Hurt, Dec 12 2015

Table[2^DivisorSigma[0, n] - 1, {n, 73}] (* Michael De Vlieger, Dec 11 2015 *)

a(n) = 2^(numdiv(n)) - 1; \\ Michel Marcus, Dec 15 2013

a(n) = -1 + 2^tau(n), where tau(n) = DivisorSigma(0, n) = A000005(n).

A243305 a(n) = 2^phi(n)+1 = A066781(n)+1.

Original entry on oeis.org

3, 3, 5, 5, 17, 5, 65, 17, 65, 17, 1025, 17, 4097, 65, 257, 257, 65537, 65, 262145, 257, 4097, 1025, 4194305, 257, 1048577, 4097, 262145, 4097, 268435457, 257, 1073741825, 65537, 1048577, 65537, 16777217, 4097, 68719476737, 262145, 16777217, 65537, 1099511627777

Offset: 1

Vincenzo Librandi, Jun 04 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A066781, A100371.

Magma
```
[2^EulerPhi(n)+1: n in [1..41]];
```

with(numtheory); A243305:=n->2^phi(n)+1; seq(A243305(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014

Mathematica
```
Table[2^EulerPhi[n] + 1, {n, 1, 50}]
```

a(n) = 2^A000010(n) + 1. - Wesley Ivan Hurt, Jun 04 2014

A243306 2^phi(n) - phi(n).

Original entry on oeis.org

1, 1, 2, 2, 12, 2, 58, 12, 58, 12, 1014, 12, 4084, 58, 248, 248, 65520, 58, 262126, 248, 4084, 1014, 4194282, 248, 1048556, 4084, 262126, 4084, 268435428, 248, 1073741794, 65520, 1048556, 65520, 16777192, 4084, 68719476700, 262126, 16777192, 65520

Offset: 1

Vincenzo Librandi, Jun 04 2014

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A066781, A100371.

[2^EulerPhi(n)-EulerPhi(n): n in [1..40]];

with(numtheory); A243306:=n->2^phi(n)-phi(n); seq(A243306(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014

Table[2^EulerPhi[n] - EulerPhi[n], {n, 1, 50}]
2^#-#&/@EulerPhi[Range[50]] (* Harvey P. Dale, Dec 19 2017 *)

a(n) = A066781(n) - A000010(n). - Wesley Ivan Hurt, Jun 04 2014

A243307 a(n) = 2^phi(n) + phi(n).

Original entry on oeis.org

3, 3, 6, 6, 20, 6, 70, 20, 70, 20, 1034, 20, 4108, 70, 264, 264, 65552, 70, 262162, 264, 4108, 1034, 4194326, 264, 1048596, 4108, 262162, 4108, 268435484, 264, 1073741854, 65552, 1048596, 65552, 16777240, 4108, 68719476772, 262162, 16777240, 65552

Offset: 1

Vincenzo Librandi, Jun 04 2014

nonn

			From _Muniru A Asiru_, Jan 23 2018: (Start)
phi(1) = 1 -> a(1) = 2^1 + 1 = 3.
phi(2) = 1 -> a(2) = 2^1 + 1 = 3.
phi(7) = 6 -> a(7) = 2^6 + 6 = 70.
phi(10) = 4 -> a(10) = 2^4 + 4 = 20.
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A066781, A100371.

List([1..1000], n -> 2^Phi(n) + Phi(n)); # Muniru A Asiru, Jan 23 2018

[2^EulerPhi(n)+EulerPhi(n): n in [1..40]];

with(numtheory); A243307:=n->2^phi(n)+phi(n); seq(A243307(n), n=1..50); # Wesley Ivan Hurt, Jun 04 2014

Table[2^EulerPhi[n] + EulerPhi[n], {n, 1, 50}]

a(n) = A066781(n) + A000010(n). - Wesley Ivan Hurt, Jun 04 2014

cp's OEIS Frontend

A226305 Numerator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

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A226306 Denominator of Product_{d|n} b(d)^Moebius(n/d), where b() = A100371().

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A100587 Number of nonempty subsets of divisors of n.

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A243305 a(n) = 2^phi(n)+1 = A066781(n)+1.

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A243306 2^phi(n) - phi(n).

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A243307 a(n) = 2^phi(n) + phi(n).

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