A076511 -id:A076511 - cp's OEIS frontend

A076512 Denominator of cototient(n)/totient(n).

1, 1, 2, 1, 4, 1, 6, 1, 2, 2, 10, 1, 12, 3, 8, 1, 16, 1, 18, 2, 4, 5, 22, 1, 4, 6, 2, 3, 28, 4, 30, 1, 20, 8, 24, 1, 36, 9, 8, 2, 40, 2, 42, 5, 8, 11, 46, 1, 6, 2, 32, 6, 52, 1, 8, 3, 12, 14, 58, 4, 60, 15, 4, 1, 48, 10, 66, 8, 44, 12, 70, 1, 72, 18, 8, 9, 60, 4, 78, 2, 2, 20, 82, 2, 64, 21

Offset: 1

Reinhard Zumkeller, Oct 15 2002

a(n)=1 iff n=A007694(k) for some k.
Numerator of phi(n)/n=Prod_{p|n} (1-1/p). - Franz Vrabec, Aug 26 2005
From Wolfdieter Lang, May 12 2011: (Start)
For n>=2, a(n)/A109395(n) = sum(((-1)^r)*sigma_r,r=0..M(n)) with the elementary symmetric functions (polynomials) sigma_r of the indeterminates {1/p_1,...,1/p_M(n)} if n = prod((p_j)^e(j),j=1..M(n)) where M(n)=A001221(n) and sigma_0=1.
This follows by expanding the above given product for phi(n)/n.
The n-th member of this rational sequence 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5,... is also (2/n^2)*sum(k,with 1<=k=2.
Therefore, this scaled sum depends only on the distinct prime factors of n.
See also A023896. Proof via PIE (principle of inclusion and exclusion). (End)
In the sequence of rationals r(n)=eulerphi(n)/n: 1, 1/2, 2/3, 1/2, 4/5, 1/3, 6/7, 1/2, 2/3, 2/5, 10/11, 1/3, ... one can observe that new values are obtained for squarefree indices (A005117); while for a nonsquarefree number n (A013929), r(n) = r(A007947(n)), where A007947(n) is the squarefree kernel of n. - Michel Marcus, Jul 04 2015

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A076511 (numerator of cototient(n)/totient(n)), A051953.

Phi(m)/m = k: A000079 \ {1} (k=1/2), A033845 (k=1/3), A000244 \ {1} (k=2/3), A033846 (k=2/5), A000351 \ {1} (k=4/5), A033847 (k=3/7), A033850 (k=4/7), A000420 \ {1} (k=6/7), A033848 (k=5/11), A001020 \ {1} (k=10/11), A288162 (k=6/13), A001022 \ {1} (12/13), A143207 (k=4/15), A033849 (k=8/15), A033851 (k=24/35).

[Numerator(EulerPhi(n)/n): n in [1..100]]; // Vincenzo Librandi, Jul 04 2015

Table[Denominator[(n - EulerPhi[n])/EulerPhi[n]], {n, 80}] (* Alonso del Arte, May 12 2011 *)

vector(80, n, numerator(eulerphi(n)/n)) \\ Michel Marcus, Jul 04 2015

a(n) = A000010(n)/A009195(n).

A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 5, 6, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 3, 4, 1, 2, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 2, 1, 2, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 4, 5, 6, 1, 2, 3, 7, 14, 13, 4, 11, 2, 1, 8

Offset: 1

Jamie Morken, May 13 2019

Row n has length A000010(n).
Row n > 1 has sum = n*A076512(n)/2.
First value on row(n) = A076511(n).
Last value on row(n) = A076512(n) for n > 1.
For n > 1, A109395(n) = Max(row) + Min(row).
For values x and y on row n > 1 at positions a and b on the row:
  x + y = A109395(n), where a = A000010(n) - (b-1).
For n > 2 the penultimate value on row A002110(n) is given by
  A038110(n)*A000040(n)-A060753(n).
From Charlie Neder, Jun 05 2019: (Start)
If p is a prime dividing n, then row p*n consists of p copies of row n.
Conjecture: If n is odd, then row 2n can be obtained from row n by interchanging the first and second halves. (End)

			The sequence as an irregular triangle:
  n/k 1, 2, 3, 4, ...
   1: 0
   2: 1
   3: 1, 2
   4: 1, 1
   5: 1, 2, 3, 4
   6: 2, 1
   7: 1, 2, 3, 4, 5, 6
   8: 1, 1, 1, 1
   9: 1, 2, 1, 2, 1, 2
  10: 3, 4, 1, 2
  11: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
  12: 2, 1, 2, 1
  13: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  14: 4, 5, 6, 1, 2, 3
  15: 7, 14, 13, 4, 11, 2, 1, 8
  ...
  Row sums: 0, 1, 3, 2, 10, 3, 21, 4, 9, 10, 55, 6, 78, 21, 60.
T(14,5) = A109395(14)*5 - A076512(14)*A038566(14,5) = 7*5 - 3*11 = 2.
T(210,2) = A109395(210)*2 - A076512(210)*A038566(210,2) = 35*2 - 8*11 = -18.

Jamie Morken, Table of n, a(n) for n = 1..13413 (Rows n = 1..210 of triangle, flattened)

Cf. A000010, A109395, A038566, A076511, A076512.

Flatten@ Table[With[{a = n/GCD[n, #], b = Numerator[#/n]}, MapIndexed[a First@ #2 - b #1 &, Flatten@ Position[GCD[Table[Mod[k, n], {k, n - 1}], n], 1] /. {} -> {1}]] &@ EulerPhi@ n, {n, 15}] (* Michael De Vlieger, Jun 06 2019 *)

vtot(n) = select(x->(gcd(n, x)==1), vector(n, k, k));
row(n) = my(q = eulerphi(n)/n, v = vtot(n)); vector(#v, k, denominator(q)*k - numerator(q)*v[k]); \\ Michel Marcus, May 14 2019

A307964 Irregular triangle read by rows: T(n,k) = A308121(A024556(n),k).

Original entry on oeis.org

7, 14, 13, 4, 11, 2, 1, 8, 3, 6, 5, 8, 3, 2, 5, 4, -1, 2, 1, 4, 13, 26, 19, 32, 25, 38, 31, 4, 17, 10, 23, 16, 29, 2, -5, 8, 1, 14, 7, 20, 11, 22, 33, 44, 31, 18, 29, 16, 27, 38, 1, 12, 23, 34, -3, 8, 19, 6, 17, 4, -9, 2, 13, 24, 5, 10, 7, 12, 9, 14, 11, 16, 5

Offset: 1

Jamie Morken, Jul 29 2019

The sequence gives odd squarefree composite rows n in A308121, i.e., rows 15, 21, 33, 35, 39, 51, 55, 57, 65, ... given by A024556(n).  These rows are the primitive rows of A308121.
Row n has length A000010(A024556(n)).
For row n:
  T(n, 1) = T(n, 2) / 2.
  T(n, phi(n)) - T(n, phi(n)-1) = T(n, 1).
  T(n, phi(n)/2+1) - T(n, phi(n)/2) = T(n, 1).
From Charlie Neder, Jul 30 2019: (Start)
For row n, T(n, k) + T(n, phi(n)-k) is constant for all k.
For 2 <= k < lpf(A024556(n)), T(n, k) = k*T(n, 1). (End)

			The sequence as an irregular triangle:
1:  7, 14, 13, 4, 11, 2, 1, 8;
2:  3, 6, 5, 8, 3, 2, 5, 4, -1, 2, 1, 4;
3:  13, 26, 19, 32, 25, 38, 31, 4, 17, 10, 23, 16, 29, 2, -5, 8, 1, 14, 7, 20;
4:  11, 22, 33, 44, 31, 18, 29, 16, 27, 38, 1, 12, 23, 34, -3, 8, 19, 6, 17, 4, -9, 2, 13, 24;
5:  5, 10, 7, 12, 9, 14, 11, 16, 5, 2, 7, 4, 9, 6, 11, 8, -3, 2, -1, 4, 1, 6, 3, 8
6:  19, 38, 25, 44, 31, 50, 37, 56, 43, 62, 49, 4, 23, 10, 29, 16, 35, 22, 41, 28, 47, 2, -11, 8, -5, 14, 1, 20, 7, 26, 13, 32;
7:  3, 6, 9, 12, 7, 10, 13, 16, 3, 6, 9, 4, 7, 10, 13, 8, 3, 6, 1, 4, 7, 10, 5, 8, 3, -2, 1, 4, 7, 2, 5, 8, -5, -2, 1, 4, -1, 2, 5, 8;
8:  7, 14, 9, 16, 11, 18, 13, 20, 15, 22, 17, 24, 7, 2, 9, 4, 11, 6, 13, 8, 15, 10, 17, 12, -5, 2, -3, 4, -1, 6, 1, 8, 3, 10, 5, 12;
9:  17, 34, 51, 68, 37, 54, 71, 88, 57, 74, 43, 12, 29, 46, 63, 32, 49, 66, 83, 4, 21, 38, 7, 24, 41, 58, 27, 44, 61, -18, -1, 16, 33, 2, 19, 36, 53, 22, -9, 8, -23, -6, 11, 28, -3, 14, 31, 48;
  ...

Jamie Morken, Table of n, a(n) for n = 1..9768

Cf. A024556, A308121, A309497, A000010, A076511, A076512.

rowsToCheck = 340;
A024556 =
Complement[Select[Range[3, rowsToCheck, 2], SquareFreeQ],
  Prime[Range[
    PrimePi[rowsToCheck]]]]; (* after Harvey P. Dale , Jan 26 2011 *)
A308121 =
Table[With[{a = n/GCD[n, #], b = Numerator[#/n]},
     MapIndexed[a First@#2 - b #1 &,
      Flatten@Position[GCD[Table[Mod[k, n], {k, n - 1}], n],
         1] /. {} -> {1}]] &@EulerPhi@n, {n,
   rowsToCheck}]; (* after Michael De Vlieger, Jun 06 2019 *)
A307964 = {};
For[i = 1, i <= Length[A024556], i++,
AppendTo[A307964, A308121[[A024556[[i]]]]]]
A307964flattened = Flatten[A307964]
(* Jamie Morken, Apr 20 2021 *)

A107473 Sum of numerator and denominator of Product_{p|n, p prime} (1 - 1/p).

Original entry on oeis.org

2, 3, 5, 3, 9, 4, 13, 3, 5, 7, 21, 4, 25, 10, 23, 3, 33, 4, 37, 7, 11, 16, 45, 4, 9, 19, 5, 10, 57, 19, 61, 3, 53, 25, 59, 4, 73, 28, 21, 7, 81, 9, 85, 16, 23, 34, 93, 4, 13, 7, 83, 19, 105, 4, 19, 10, 31, 43, 117, 19, 121, 46, 11, 3, 113, 43, 133, 25, 113, 47, 141, 4, 145, 55, 23, 28

Offset: 1

Leroy Quet, May 27 2005

nonn

			a(12) = 4 = 1+3 because (1 - 1/2)(1 - 1/3) = 1/3.

Cf. A076511, A076512.

a:=proc(n) local b,ct,f; with(numtheory): b:=convert(factorset(n),list): ct:=nops(b): f:=simplify(product(1-1/b[j],j=1..ct)):numer(f)+denom(f) end: seq(a(n),n=1..100); # Emeric Deutsch, May 28 2005

a(n) = A076511(n) + A076512(n). - Michel Marcus, Aug 16 2019

More terms from Emeric Deutsch, May 28 2005

cp's OEIS Frontend

A076512 Denominator of cototient(n)/totient(n).

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A308121 Irregular triangle read by rows: T(n,k) = A109395(n)*k-A076512(n)*A038566(n,k).

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A307964 Irregular triangle read by rows: T(n,k) = A308121(A024556(n),k).

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A107473 Sum of numerator and denominator of Product_{p|n, p prime} (1 - 1/p).

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A308121 Irregular triangle read by rows: T(n,k) = A109395(n)k-A076512(n)A038566(n,k).