A066781 -id:A066781 - cp's OEIS frontend

A100371 a(n) = 2^phi(n) - 1 = A066781(n) - 1.

1, 1, 3, 3, 15, 3, 63, 15, 63, 15, 1023, 15, 4095, 63, 255, 255, 65535, 63, 262143, 255, 4095, 1023, 4194303, 255, 1048575, 4095, 262143, 4095, 268435455, 255, 1073741823, 65535, 1048575, 65535, 16777215, 4095, 68719476735, 262143, 16777215, 65535

Offset: 1

Labos Elemer, Nov 30 2004

Number of nonempty subsets of reduced residue system [RRS(n)] modulo n.

T. D. Noe, 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. A000010, A066781.

A100371:=n->2^numtheory[phi](n)-1: seq(A100371(n), n=1..60); # Wesley Ivan Hurt, Apr 14 2017

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

a(n) = 2^eulerphi(n) - 1; \\ Michel Marcus, Apr 14 2017

from sympy import totient
def a(n): return 2**totient(n) - 1 # Indranil Ghosh, Apr 14 2017

a(n) = Sum_{i=1..n} binomial(phi(n), i). - Enrique Pérez Herrero, Mar 10 2012

Entry revised by N. J. A. Sloane, Jun 07 2013

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

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).

A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).

Original entry on oeis.org

1, 12, 40, 48, 60, 192, 544, 640, 680, 704, 768, 816, 960, 1020, 1664, 3072, 10240, 11008, 12288, 13760, 15360, 19456, 24320, 49152, 83968, 125952, 131584, 139264, 139808, 163840, 164480, 174080, 174760, 196608, 197376, 208896, 209712, 245760, 246720, 261120, 262140, 720896, 786432

Offset: 1

Max Alekseyev and Altug Alkan, Sep 18 2017

Conjecture: For n > 1, a(n) is a Zumkeller number (A083207) [confirmed for n up to 47]. - Ivan N. Ianakiev, Sep 22 2017

			704 = 11*2^6 is a term since phi(11*2^6) = 5*2^6 and 11*2^6 divides 2^(5*2^6) - 5*2^6.

Giovanni Resta, Table of n, a(n) for n = 1..180 (terms < 10^12; first 101 terms from Michel Marcus)

Cf. A000010, A007733, A066781.

{1}~Join~Select[Range[10^6], Function[n, # == PowerMod[2, #, n] &@ EulerPhi@ n]] (* Michael De Vlieger, Sep 18 2017 *)

isok(n) = Mod(2, n)^eulerphi(n)==eulerphi(n);

Let m be an odd number, z = A007733(m) and k, 0 <= k < z, be such that phi(m) == 2^k (mod m); then m*2^(i*z - k + 1) belongs to this sequence for all i >= 1. And this is a general form of the terms of this sequence.
Some families of solutions of the form m*2^(i*z - k + 1):
If m = 3, then z = 2 and k = 1 ==> 3*2^(2*i) is a term for all i >= 1.
If m = 5, then z = 4 and k = 2 ==> 5*2^(4*i-1) is a term for all i >= 1.
If m = 7, then z = 3 but k does not exist ==> no term with odd part equal to 7.
If m = 15, then z = 4 and k = 3 ==> 15*2^(4*i-2) is a term for all i >= 1.
If m = 77, then z = 30 and k = 14 ==> 77*2^(30*i-13) is a term for all i >= 1.

A073318 a(n) = 2^phi(n) - Sum_{j=0..n} binomial(phi(n), phi(j)).

Original entry on oeis.org

0, -1, -2, -3, -6, -4, -10, -13, -26, -14, -183, -15, -22, -57, -210, -211, -1730, -58, 25160, -240, -3356, -949, 238031, -241, -256823, -3918, -143243, -3919, 46326924, -242, 281620682, -61817, -639769, -61818, -4718174, -4415, 2023569890, -224436, -7556927, -63639, -43279525745, -4416

Offset: 1

Labos Elemer, Jul 26 2002

a(n) > 0 for {19, 23, 29, 31, 37, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97}. Does this hold only for special primes?

No: composites for which a(n) > 0 include 121, 289, 437, 529, 667, 671, 697, 703, 713, 731, .... - Robert Israel, Jan 23 2021

Robert Israel, Table of n, a(n) for n = 1..3322

Cf. A066781, A073318.

g[x_] := EulerPhi[x] Table[Apply[Plus, Table[Binomial[g[n], g[j]], {j, 0, n}]], {n, 1, 50}]

a(n) = A066781(n) - A073317(n).

A073317 a(n)=Sum_{j = 0..n} binomial(phi(n),phi(j)).

Original entry on oeis.org

2, 3, 6, 7, 22, 8, 74, 29, 90, 30, 1207, 31, 4118, 121, 466, 467, 67266, 122, 236984, 496, 7452, 1973, 3956273, 497, 1305399, 8014, 405387, 8015, 222108532, 498, 792121142, 127353, 1688345, 127354, 21495390, 8511, 66695906846, 486580

Offset: 1

Labos Elemer, Jul 26 2002

Cf. A066781, A073318.

g[x_] := EulerPhi[x] Table[Apply[Plus, Table[Binomial[g[n], g[j]], {j, 0, n}]], {n, 1, 50}]

A189914 a(n) is 2^phi(n) times the least common multiple of the proper divisors of n.

Original entry on oeis.org

1, 2, 2, 4, 8, 16, 24, 64, 64, 192, 160, 1024, 192, 4096, 896, 3840, 2048, 65536, 1152, 262144, 5120, 86016, 22528, 4194304, 6144, 5242880, 106496, 2359296, 114688, 268435456, 7680, 1073741824, 1048576, 34603008, 2228224, 587202560, 147456, 68719476736, 9961472

Offset: 0

Peter Luschny, Jun 22 2011

nonn

The sequence relates arithmetic properties of roots of unity in the complex plane with number theoretic properties of integers. This connection often appears as intriguing identities showing products of specific values of the sine function or the gamma function reducing to simple values (see for instance the first formula below).

Charles R Greathouse IV, Table of n, a(n) for n = 0..3322
Peter Luschny and Stefan Wehmeier, The lcm(1,2,...,n) as a product of sine values sampled over the points in Farey sequences, arXiv:0909.1838 [math.CA], Sep 2009.
Albert Nijenhuis, Short Gamma Products with Simple Values, The American Mathematical Monthly, Vol. 117, No. 8, Oct 2010, pp. 733-737.
J. Sándor and L. Tóth, A remark on the gamma function, Elemente der Mathematik, 44 (1989), pp. 73-76, Birkhäuser.

A189914 := n -> 2^numtheory[phi](n)*ilcm(op(numtheory[divisors](n) minus {1,n})): seq(A189914(n), n=0..35);

a[n_] := 2^EulerPhi[n] * LCM @@ Most[Divisors[n]]; a[0] = 1; a[1] = 2; Table[a[n], {n, 0, 38}] (* Jean-François Alcover, Jan 22 2014 *)

a(n)=if(n,my(p=n); if(isprime(n)||(ispower(n, , &p)&&isprime(p)), n/p, n)<Charles R Greathouse IV, Jun 24 2011

Let R(n) = {k | gcd(n,k) = 1, k = 1..floor(n/2)} and b(n) = product_{R(n)} sin(Pi*k/n) then a(n) = n / b(n)^2 for n > 1.

a(n) = A066781(n)*A048671(n).

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

A073319 Numbers n such that A073318(n) = 2^phi(n) - Sum_{j=0..n} binomial(phi(n), phi(j)) is positive.

Original entry on oeis.org

19, 23, 29, 31, 37, 43, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 121, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 289, 293

Offset: 1

Labos Elemer, Jul 26 2002

nonn

			Several values are composites: 121, 289, 437, 529, ..., 961, 989. Primes like 2, ..., 17, 41 are not here.

Cf. A066781, A073317, A073318.

g[x_] := EulerPhi[x] Do[s=2^g[n]-Apply[Plus, Table[Binomial[g[n], g[j]], {j, 0, n}]]; If[Sign[s]==1&&!PrimeQ[n], k=k+1; Print[{k, n, PrimeQ[n]}]], {n, 1, 1000}]

Solutions to A066781(x) - A073317(x) > 0.

cp's OEIS Frontend

A100371 a(n) = 2^phi(n) - 1 = A066781(n) - 1.

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

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A100587 Number of nonempty subsets of divisors of n.

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A292544 Numbers h such that 2^phi(h) == phi(h) (mod h).

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A073318 a(n) = 2^phi(n) - Sum_{j=0..n} binomial(phi(n), phi(j)).

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A073317 a(n)=Sum_{j = 0..n} binomial(phi(n),phi(j)).

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A189914 a(n) is 2^phi(n) times the least common multiple of the proper divisors of n.

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